A Ф-local finite group consists of a finite Ф-group S, together with a ...

EXTENSIONS OF p-LOCAL FINITE GROUPS

C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, AND B. OLIVER

Abstract. A p-local �nite group consists of a �nite p-group S, together with a pairof categories which encode \conjugacy" relations among subgroups of S, and whichare modelled on the fusion in a Sylow p-subgroup of a �nite group. It contains enoughinformation to de�ne a classifying space which has many of the same properties asp-completed classifying spaces of �nite groups. In this paper, we study and classifyextensions of p-local �nite groups, and also compute the fundamental group of theclassifying space of a p-local �nite group.

A p-local �nite group consists of a �nite p-group S, together with a pair of categories(F ;L), of which F is modeled on the conjugacy (or fusion) in a Sylow subgroup of a�nite group. The category L is essentially an extension of F and contains just enoughextra information so that its p-completed nerve has many of the same properties asp-completed classifying spaces of �nite groups. We recall the precise de�nitions of theseobjects in Section 1, and refer to [BLO2] and [5A1] for motivation for their study.

In this paper, we study extensions of saturated fusion systems and of p-local �nitegroups. This is in continuation of our more general program of trying to understand towhat extent properties of �nite groups can be extended to properties of p-local �nitegroups, and to shed light on the question of how many (exotic) p-local �nite groupsthere are. While we do not get a completely general theory of extensions of one p-local�nite group by another, we do show how certain types of extensions can be describedin manner very similar to the situation for �nite groups.

From the point of view of group theory, developing an extension theory for p-local�nite groups is related to the question of to what extent the extension problem forgroups is a local problem, i.e., a problem purely described in terms of a Sylow p-subgroup and conjugacy relations inside it. In complete generality this is not thecase. For example, strongly closed subgroups of a Sylow p-subgroup S of G need notcorrespond to normal subgroups of G. However, special cases where this does happeninclude the case of existence of p-group quotients (the focal subgroup theorems, see[Go, xx7.3{7.4]) and central subgroups (described via the Z�-theorem of Glauberman[Gl]).

From the point of view of homotopy theory, one of the problems which comes upwhen looking for a general theory of extensions of p-local �nite groups is that whilean extension of groups 1 ! K ! � ! G ! 1 always induces a (homotopy) �brationsequence of classifying spaces, it does not in general induce a �bration sequence ofp-completed classifying spaces. Two cases where this does happen are those where

1991 Mathematics Subject Classi�cation. Primary 55R35. Secondary 55R40, 20D20.Key words and phrases. Classifying space, p-completion, �nite groups, fusion.C. Broto is partially supported by MCYT grant BFM2001{2035.N. Castellana is partially supported by MCYT grant BFM2001{2035.J. Grodal is partially supported by NSF grants DMS-0104318 and DMS-0354633.R. Levi is partially supported by EPSRC grant GR/M7831.B. Oliver is partially supported by UMR 7539 of the CNRS.

1

2 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER

G is a p-group, and where the extension is central. In both of these cases, BH^p is

the homotopy �ber of the map B�^p ��! BG^p . Thus, also from the point of view

of homotopy theory, it is natural to study extensions of p-local �nite groups with p-group quotient, and to study central extensions of p-local �nite groups. The thirdcase we study is that of extensions with quotient of order prime to p; and the caseof p and p0-group quotients to some extent unify to give a theory of extensions withp-solvable quotient. Recall in this connection that by a previous result of ours [5A1,Proposition C], solvable p-local �nite groups all come from p-solvable groups. In allthree of these situations, we develop a theory of extensions which parallels the situationfor �nite groups.

We now describe the contents of this paper in more detail, stating simpli�ed versionsof our main results on extensions. Stronger and more precise versions of some of thesetheorems will be stated and proven later.

In Section 3, we construct a very general theory of fusion subsystems (Proposition3.8) and linking subsystems (Theorem 3.9) with quotient a p-group or a group of orderprime to p. As a result we get the following theorem (Corollary 3.10), which for ap-local �nite group (S;F ;L), describes a correspondence between covering spaces ofthe geometric realization jLj and certain p-local �nite subgroups of (S;F ;L).

Theorem A. Suppose that (S;F ;L) is a p-local �nite group. Then there is a normalsubgroup H C �1(jLj) which is minimal among all those whose quotient is �nite and p-solvable. Any covering space of the geometric realization jLj whose fundamental groupcontains H is homotopy equivalent to jL0j for some p-local �nite group (S 0;F 0;L0),where S 0 � S and F 0 � F .

Moreover, the p-local �nite group (S 0;F 0;L0) of Theorem A can be explicitly de-scribed in terms of L, as we will explain in Section 3.

In order to use this theorem, it is useful to have ways of �nding the �nite p-solvablequotients of �1(jLj). This can be done by iteration, using the next two theorems. Inthem, the maximal p-group quotient of �1(jLj), and the maximal quotient of orderprime to p, are described solely in terms of the fusion system F .

When G is an in�nite group, we de�ne Op(G) and Op0(G) to be the intersectionof all normal subgroups of G of p-power index, or index prime to p, respectively.These clearly generalize the usual de�nitions for �nite G (but are not the only possiblegeneralizations).

Theorem B (Hyperfocal subgroup theorem for p-local �nite groups). For a p-local�nite group (S;F ;L), the natural homomorphism

S ��! �1(jLj)=Op(�1(jLj)) �= �1(jLj

^p )

is surjective, with kernel equal to

OpF(S)

def=g�1�(g) 2 S

�� g 2 P � S; � 2 Op(AutF(P ))�:

For any saturated fusion system F over a p-group S, we let Op0

� (F) � F be thesmallest fusion subsystem of F (in the sense of De�nition 1.1) which contains allautomorphism groups Op0(AutF(P )) for P � S. Equivalently, Op0

� (F) is the smallestsubcategory of F with the same objects, and which contains all restrictions of allautomorphisms in F of p-power order. This subcategory is needed in the statement ofthe next theorem.

EXTENSIONS OF P -LOCAL FINITE GROUPS 3

Theorem C. For a p-local �nite group (S;F ;L), the natural map

OutF(S) ��! �1(jLj)=Op0(�1(jLj))

is surjective, with kernel equal to

Out0F(S)def=� 2 OutF(S)

��jP 2 MorOp

0

� (F)(P; S); some F -centric P � S

�:

Theorem B is proved as Theorem 2.5, and Theorem C is proved as Theorem 5.5.

In fact, we give a purely algebraic description of these subsystems of \p-power index"or of \index prime to p" (De�nition 3.1), and then show in Sections 4.1 and 5.1 thatthey in fact all arise as �nite covering spaces of jLj (see Theorems 4.4 and 5.5). Similarresults were also discovered independently by Puig [Pu4]. Then, in Sections 4.2 and5.2, we establish converses to these concerning the extensions of a p-local �nite group,which include the following theorem.

Theorem D. Let (S;F ;L) be a p-local �nite group. Suppose we are given a �brationsequence jLj^p ! E ! BG, where G is a �nite p-group or has order prime to p.Then there exists a p-local �nite group (S 0;F 0;L0) such that jL0j^p ' E^

p .

This is shown as Theorems 4.7 and 5.8. Moreover, when G is a p-group, we give inTheorem 4.7 an explicit algebraic construction of the p-local �nite group (S 0;F 0;L0).

Finally, in Section 6, we develop the theory of central extensions of p-local �nitegroups. Our main results there (Theorems 6.8 and 6.13) give a more elaborate versionof the following theorem. Here, the center of a p-local �nite group (S;F ;L) is de�nedto be the subgroup of elements x 2 Z(S) such that �(x) = x for all � 2 Mor(F c).

Theorem E. Suppose that A is a central subgroup of a p-local �nite group (S;F ;L).Then there exists a canonical quotient p-local �nite group (S=A;F=A;L=A), and thecanonical projection of jLj onto jL=Aj is a principal �bration with �ber BA.

Conversely, for any principal �bration E ! jLj with �ber BA, where A is a �nite

abelian p-group, there exists a p-local �nite group (eS; eF ; eL) such that j eLj ' E. Further-more, this correspondence sets up a 1� 1 correspondence between central extensions ofL by A and elements in H2(jLj;A).

One motivation for this study was the question of whether extensions of p-local �nitegroups coming from �nite groups can produce exotic p-local �nite groups. In the caseof central extensions, we are able to show that (S;F ;L) comes from a �nite groupif and only if (S=A;F=A;L=A) comes from a group (Corollary 6.14). For the othertypes of extensions studied in this paper, this is still an open question. We have sofar failed to produce exotic examples in this way, and yet we have also been unable toshow that exotic examples cannot occur. This question seems to be related to somerather subtle and interesting group theoretic issues relating local to global structure;see Corollary 4.8 for one partial result in this direction.

This paper builds on the earlier paper [5A1] by the same authors, and many of theresults in that paper were originally motivated by this work on extensions.

The authors would like to thank the University of Aberdeen, Universitat Aut�onomade Barcelona, Universit�e Paris 13, and Aarhus Universitet for their hospitality. Inparticular, the origin of this project goes back to a three week period in the spring of2001, when four of the authors met in Aberdeen.


1. A quick review of p-local finite groups

We �rst recall the de�nitions of a fusion system, and a saturated fusion system, inthe form given in [BLO2]. For any group G, and any pair of subgroups H;K � G, weset

NG(H;K) = fx 2 G j xHx�1 � Kg;

let cx denote conjugation by x (cx(g) = xgx�1), and set

HomG(H;K) =�cx 2 Hom(H;K)

��x 2 NG(H;K)�= NG(H;K)=CG(H):

By analogy, we also write

AutG(H) = HomG(H;H) =�cx 2 Aut(H)

��x 2 NG(H)�= NG(H)=CG(H):

De�nition 1.1 ([Pu2] and [BLO2, De�nition 1.1]). A fusion system over a �nite p-group S is a category F , where Ob(F) is the set of all subgroups of S, and whichsatis�es the following two properties for all P;Q � S:

� HomS(P;Q) � HomF(P;Q) � Inj(P;Q); and

� each ' 2 HomF(P;Q) is the composite of an isomorphism in F followed by aninclusion.

The following additional de�nitions and conditions are needed in order for thesesystems to be very useful. If F is a fusion system over a �nite p-subgroup S, then twosubgroups P;Q � S are said to be F-conjugate if they are isomorphic as objects of thecategory F .

De�nition 1.2 ([Pu2], see [BLO2, Def. 1.2]). Let F be a fusion system over a p-groupS.

� A subgroup P � S is fully centralized in F if jCS(P )j � jCS(P 0)j for all P 0 � Swhich is F-conjugate to P .

� A subgroup P � S is fully normalized in F if jNS(P )j � jNS(P0)j for all P 0 � S

which is F-conjugate to P .

� F is a saturated fusion system if the following two conditions hold:

(I) For all P � S which is fully normalized in F , P is fully centralized in F andAutS(P ) 2 Sylp(AutF(P )).

(II) If P � S and ' 2 HomF(P; S) are such that 'P is fully centralized, and if weset

N' = fg 2 NS(P ) j'cg'�1 2 AutS('P )g;

then there is ' 2 HomF (N'; S) such that 'jP = '.

If G is a �nite group and S 2 Sylp(G), then by [BLO2, Proposition 1.3], the categoryFS(G) de�ned by letting Ob(FS(G)) be the set of all subgroups of S and settingMorFS(G)(P;Q) = HomG(P;Q) is a saturated fusion system.

An alternative pair of axioms for a fusion system being saturated have been givenby Radu Stancu [St]. He showed that axioms (I) and (II) above are equivalent to thetwo axioms:

(I0) Inn(S) 2 Sylp(AutF(S)).


(II0) If P � S and ' 2 HomF(P; S) are such that 'P is fully normalized, and if we set

N' = fg 2 NS(P ) j'cg'�1 2 AutS('P )g;

then there is ' 2 HomF (N'; S) such that 'jP = '.

The following consequence of conditions (I) and (II) above will be needed severaltimes throughout the paper.

Lemma 1.3. Let F be a saturated fusion system over a p-group S. Let P; P 0 � S bea pair of F-conjugate subgroups such that P 0 is fully normalized in F . Then there is ahomomorphism � 2 HomF (NS(P ); NS(P

0)) such that �(P ) = P 0.

Proof. This is shown in [BLO2, Proposition A.2(b)]. �

In this paper, it will sometimes be necessary to work with fusion systems which arenot saturated. This is why we have emphasized the di�erence between fusion systems,and saturated fusion systems, in the above de�nitions.

We next specify certain collections of subgroups relative to a given fusion system.

De�nition 1.4. Let F be a fusion system over a �nite p-subgroup S.

� A subgroup P � S is F -centric if CS(P0) = Z(P 0) for all P 0 � S which is F-

conjugate to P .

� A subgroup P � S is F -radical if OutF (P ) is p-reduced; i.e., if Op(OutF(P )) = 1.

� For any P � S which is fully centralized in F , the centralizer fusion system CF(P )is the fusion system over CS(P ) de�ned by setting

HomCF (P )(Q;Q0) =

��jQ

�� 2 HomF(QP;Q0P ); �jP = IdP ; �(Q) � Q0

:

A subgroup P � S is F -quasicentric if for all P 0 � S which is F-conjugate to P andfully centralized in F , CF(P

0) is the fusion system of the p-group CS(P0).

� F c � F q � F denote the full subcategories of F whose objects are the F-centricsubgroups, and F-quasicentric subgroups, respectively, of S.

If F = FS(G) for some �nite group G, then P � S is F -centric if and only ifP is p-centric in G (i.e., Z(P ) 2 Sylp(CG(P ))), and P is F -radical if and only ifNG(P )=(P �CG(P )) is p-reduced. Also, P is F -quasicentric if and only if CG(P ) containsa normal subgroup of order prime to p and of p-power index.

In fact, when working with p-local �nite groups, it suÆces to have a fusion systemF c de�ned on the centric subgroups of S, and which satis�es axioms (I) and (II) abovefor those centric subgroups. In other words, fusion systems de�ned only on the centricsubgroups are equivalent to fusion systems de�ned on all subgroups, as described inthe following theorem.

Theorem 1.5. Fix a p-group S and a fusion system F over S.

(a) Assume F is saturated. Then each morphism in F is a composite of restrictionsof morphisms between subgroups of S which are F-centric, F-radical, and fullynormalized in F . More precisely, for each P; P 0 � S and each ' 2 IsoF (P; P

0), thereare subgroups P = P0; P1; : : : ; Pk = P 0, subgroups Qi � hPi�1; Pii (i = 1; : : : ; k)which are F-centric, F-radical, and fully normalized in F , and automorphisms'i 2 AutF(Qi), such that 'i(Pi�1) = Pi for all i and ' = 'k Æ � � � Æ '1jP .


(b) Assume conditions (I) and (II) in De�nition 1.2 are satis�ed for all F-centric sub-groups P � S. Assume also that each morphism in F is a composite of restrictionsof morphisms between F-centric subgroups of S. Then F is saturated.

Proof. Part (a) is Alperin's fusion theorem for saturated fusion systems, in the formshown in [BLO2, Theorem A.10]. Part (b) is a special case of [5A1, Theorem 2.2]: thecase where H is the set of all F -centric subgroups of S. �

Theorem 1.5(a) will be used repeatedly throughout this paper. The following lemmais a �rst easy application of the theorem, and provides a very useful criterion for asubgroup to be quasicentric or not.

Lemma 1.6. Let F be a saturated fusion system over a p-group S. Then the followinghold for any P � S.

(a) Assume that P � Q � P �CS(P ) and Id 6= � 2 AutF(Q) are such that �jP = IdPand � has order prime to p. Then P is not F-quasicentric.

(b) Assume that P is fully centralized in F , and is not F-quasicentric. Then there areP � Q � P �CS(P ) and Id 6= � 2 AutF(Q) such that Q is F-centric, �jP = IdP ,and � has order prime to p.

Proof. (a) Fix any P 0 which is F -conjugate to P and fully centralized in F . By axiom(II), there is ' 2 HomF(Q; S) such that '(P ) = P 0; set Q0 = '(Q). Thus '�'�1jCQ0 (P 0)is an automorphism in CF(P

0) whose order is not a power of p, so CF(P0) is not the

fusion system of CS(P0), and P is not F -quasicentric.

(b) Assume that P is fully centralized in F and not F -quasicentric. Then CF(P )strictly contains the fusion system of CS(P ) (since CF(P

0) is isomorphic as a categoryto CF(P ) for all P

0 which is F -conjugate to P and fully centralized in F). Since CF(P )is saturated by [BLO2, Proposition A.6], Theorem 1.5(a) implies there is a subgroupQ � CS(P ) which is CF(P )-centric and fully normalized in CF(P ), and such thatAutCF (P )(Q) AutCS(P )(Q). Since Q is fully normalized, AutCS(P )(Q) is a Sylow p-subgroup of AutCF (P )(Q), and hence this last group is not a p-group. Also, by [BLO2,Proposition 2.5(a)], PQ is F -centric since Q is CF(P )-centric. �

Since orbit categories | both of fusion systems and of groups | will play a role incertain proofs in the last three sections, we de�ne them here.

De�nition 1.7. (a) If F is a fusion system over a p-group S, then Oc(F) (the centricorbit category of F) is the category whose objects are the F-centric subgroups ofS, and whose morphism sets are given by

MorOc(F)(P;Q) = RepF(P;Q)def= QnHomF(P;Q):

Let ZF : Oc(F) ! Z(p)-mod be the functor which sends P to Z(P ) and ['] (the

class of ' 2 HomF(P;Q)) to Z(Q)'�1

��! Z(P ).

(b) If G is a �nite group and S 2 Sylp(G), then OcS(G) (the centric orbit category of

G) is the category whose objects are the subgroups of S which are p-centric in G,and where

MorOcS(G)

(P;Q) = QnNG(P;Q) �= MapG(G=P;G=Q):

Let ZG : OcS(G) ! Z(p)-mod be the functor which sends P to Z(P ) and [g] (the

class of g 2 NG(P;Q)) to conjugation by g�1.


We now turn to linking systems associated to abstract fusion systems.

De�nition 1.8 ([BLO2, Def. 1.7]). Let F be a fusion system over the p-group S.A centric linking system associated to F is a category L whose objects are the F-centric subgroups of S, together with a functor � : L ��! F c, and \distinguished"

monomorphisms PÆP��! AutL(P ) for each F-centric subgroup P � S, which satisfy

the following conditions.

(A) � is the identity on objects. For each pair of objects P;Q 2 L, Z(P ) acts freelyon MorL(P;Q) by composition (upon identifying Z(P ) with ÆP (Z(P )) � AutL(P )),and � induces a bijection

MorL(P;Q)=Z(P )�=

��! HomF(P;Q):

(B) For each F-centric subgroup P � S and each x 2 P , �(ÆP (x)) = cx 2 AutF(P ).

(C) For each f 2 MorL(P;Q) and each x 2 P , the following square commutes in L:

Pf! Q

P

ÆP (x)#

f!Q:

ÆQ(�(f)(x))#

A p-local �nite group is de�ned to be a triple (S;F ;L), where S is a �nite p-group,F is a saturated fusion system over S, and L is a centric linking system associated toF . The classifying space of the triple (S;F ;L) is the p-completed nerve jLj^p .

For any �nite group G with Sylow p-subgroup S, a category LcS(G) was de�ned in[BLO1], whose objects are the p-centric subgroups of G, and whose morphism sets arede�ned by

MorLcS(G)(P;Q) = NG(P;Q)=Op(CG(P )):

Since CG(P ) = Z(P ) � Op(CG(P )) when P is p-centric in G, LcS(G) is easily seento satisfy conditions (A), (B), and (C) above, and hence is a centric linking systemassociated to FS(G). Thus (S;FS(G);LcS(G)) is a p-local �nite group, with classifyingspace jLcS(G)j

^p ' BG^

p (see [BLO1, Proposition 1.1]).

It will be of crucial importance in this paper that any centric linking system asso-ciated to F can be extended to a quasicentric linking system; a linking system withsimilar properties, whose objects are the F -quasicentric subgroups of S. We �rst makemore precise what this means.

De�nition 1.9. Let F be any saturated fusion system over a p-group S. A quasicentriclinking system associated to F consists of a category Lq whose objects are the F-quasicentric subgroups of S, together with a functor � : Lq ��! F q, and distinguishedmonomorphisms

P �CS(P )ÆP��! AutLq(P );

which satisfy the following conditions.

(A)q � is the identity on objects and surjective on morphisms. For each pair of objectsP;Q 2 Lq such that P is fully centralized, CS(P ) acts freely on MorLq(P;Q) bycomposition (upon identifying CS(P ) with ÆP (CS(P )) � AutLq(P )), and � inducesa bijection

MorLq(P;Q)=CS(P )�=

��! HomF (P;Q):


(B)q For each F-quasicentric subgroup P � S and each g 2 P , � sends ÆP (g) 2AutLq(P ) to cg 2 AutF(P ).

(C)q For each f 2 MorLq(P;Q) and each x 2 P , f Æ ÆP (x) = ÆQ(�(f)(x)) Æ f inMorLq(P;Q).

(D)q For each F-quasicentric subgroup P � S, there is a morphism �P 2 MorLq(P; S)such that �(�P ) = inclSP 2 Hom(P; S), and such that for each g 2 P �CS(P ), ÆS(g) Æ�P = �P Æ ÆP (g) in MorLq(P; S).

If P and P 0 are F -conjugate and F -quasicentric, then for any Q � S, MorLq(P;Q) �=MorLq(P

0; Q) and HomF(P;Q) �= HomF(P0; Q), while the centralizers CS(P ) and

CS(P0) need not have the same order. This is why condition (A)q makes sense only if

we assume that P is fully centralized; i.e., that CS(P ) is as large as possible. WhenP is F -centric, then this condition is irrelevant, since every subgroup P 0 which isF -conjugate to P is fully centralized (CS(P

0) = Z(P 0) �= Z(P )).

Note that (D)q is a special case of (C)q when P is F -centric; this is why the axiomis not needed for centric linking systems. We also note the following relation betweenthese axioms:

Lemma 1.10. In the situation of De�nition 1.9, axiom (C)q implies axiom (B)q.

Proof. Fix an F -quasicentric subgroup P � S, and an element g 2 P . We apply (C)qwith f = ÆP (g). For each x 2 P , if we set y = �(ÆP (g))(x), then ÆP (g) Æ ÆP (x) =ÆP (y) Æ ÆP (g). Since ÆP is an injective homomorphism, this implies that gx = yg, andthus that y = cg(x). So �(ÆP (g)) = cg. �

When (S;F ;L) is a p-local �nite group, and Lq is a quasicentric linking systemassociated to F , then we say that Lq extends L if the full subcategory of Lq withobjects the F -centric subgroups of S is isomorphic to L via a functor which commuteswith the projection functors to F and with the distinguished monomorphisms. In[5A1, Propositions 3.4 & 3.12], we constructed an explicit quasicentric linking systemLq associated to F and extending L, and showed that it is unique up to an isomorphismof categories which preserves all of these structures. So from now on, we will simplyrefer to Lq as the quasicentric linking system associated to (S;F ;L).

Condition (D)q above helps to motivate the following de�nition of inclusion mor-phisms in a quasicentric linking system.

De�nition 1.11. Fix a p-local �nite group (S;F ;L), with associated quasicentric link-ing system Lq.

(a) A morphism �P 2 MorLq(P; S) is an inclusion morphim if it satis�es the hypothesesof axiom (D)q: if �(�P ) = inclSP , and if ÆS(g) Æ �P = �P Æ ÆP (g) in MorLq(P; S) forall g 2 P �CS(P ). If P = S, then we also require that �S = IdS.

(b) A compatible set of inclusions for Lq is a choice of morphisms f�QP g for all pairs of

F-quasicentric subgroups P � Q, such that �QP 2 MorLq(P;Q), such that �RP = �RQ Æ�QP

for all P � Q � R, and such that �SP is an inclusion morphism for each P .

The following properties of quasicentric linking systems were also proven in [5A1].

Proposition 1.12. The following hold for any p-local �nite group (S;F ;L), with as-sociated quasicentric linking system Lq.


(a) The inclusion L � Lq induces a homotopy equivalence jLj ' jLqj between geometricrealizations. More generally, for any full subcategory L0 � Lq which contains asobjects all subgroups of S which are F-centric and F-radical, the inclusion L0 � Lq

induces a homotopy equivalence jL0j ' jLqj.

(b) Let ' 2 MorLq(P;R) and 2 MorLq(Q;R) be any pair of morphisms in Lq withthe same target group such that Im(�(')) � Im(�( )). Then there is a uniquemorphism � 2 MorLq(P;Q) such that ' = Æ �.

Proof. The homotopy equivalences jLqj ' jLj ' jL0j are shown in [5A1, Theorem 3.5].Point (b) is shown in [5A1, Lemma 3.6]. �

Point (b) above will be frequently used throughout the paper. In particular, it makesit possible to embed the linking system of S (or an appropriate full subcategory) inLq, depending on the choice of an inclusion morphisms �P as de�ned above, for eachobject P . Such inclusion morphisms always exist by axiom (D)q. In the followingproposition, LS(S)jOb(Lq) denotes the full subcategory of LS(S) whose objects are theF -quasicentric subgroups of S.

In general, for a functor F : C ! C 0, and objects c; d 2 Ob(C), we let Fc;d denotethe map from MorC(c; d) to MorC0(F (c); F (d)) induced by F .

Proposition 1.13. Fix a p-local �nite group (S;F ;L). let Lq be its associated qua-sicentric linking system, and let � : Lq ��! F q be the projection. Then any choice ofinclusion morphisms �P = �SP 2 MorLq(P; S), for all F-quasicentric subgroups P � S,extends to a unique inclusion of categories

Æ : LS(S)jOb(Lq) ��! Lq

such that ÆP;S(1) = �P for all P ; and such that

(a) ÆP;P (g) = ÆP (g) for all g 2 P � CS(P ), and

(b) �(ÆP;Q(g)) = cg 2 Hom(P;Q) for all g 2 NS(P;Q).

In addition, the following hold.

(c) If we set �QP = ÆP;Q(1) for all P � Q, then f�QP g is a compatible set of inclusionsfor Lq.

(d) For any P C Q � S, where P and Q are both F-quasicentric and P is fullycentralized in F , and any morphism 2 AutLq(P ) which normalizes ÆP;P (Q),

there is a unique b 2 AutLq(Q) such that b Æ �QP = �QP Æ . Furthermore, for any

g 2 Q, ÆP;P (g) �1 = ÆP;P (�( b )(g)).

(e) Every morphism ' 2 MorLq(P;Q) in Lq is a composite ' = �QP 0 Æ '

0 for a uniquemorphism '0 2 IsoLq(P; P

0), where P 0 = Im(�(')).

Proof. For each P and Q, and each g 2 NS(P;Q), there is by Proposition 1.12(b) aunique morphism ÆP;Q(g) such that

ÆS(g) Æ �P = �Q Æ ÆP;Q(g):

This de�nes Æ on morphism sets, and also allows us to de�ne �QP = ÆP;Q(1). Then by

the axioms in De�nition 1.9, f�QP g is a compatible set of inclusions for Lq, and Æ isa functor which satis�es (a), (b), and (c). Point (e) is a special case of Proposition1.12(b) (where P 0 = Im(�('))).


If ÆP;Q(g) = ÆP;Q(g0) for g; g0 2 NG(P;Q), then ÆS(g) Æ �P = ÆS(g

0) Æ �P , and henceg = g0 by [5A1, Lemma 3.9]. Thus each ÆP;Q is injective.

It remains to prove (d). Set ' = �( ) 2 AutF (P ) for short. Since normalizesÆP;P (Q), for all g 2 Q there is h 2 Q such that ÆP;P (g)

�1 = ÆP;P (h), and this impliesthe relation 'cg'

�1 = ch in AutF(P ). Thus Q is contained in N'. So by axiom (II)(and since P is fully centralized), ' extends to ' 2 IsoF(Q;Q

0) for some P C Q0 � S.

Let b 0 2 IsoLq(Q;Q0) be any lifting of ' to Lq.

By axiom (A)q (and since is an isomorphism), there is x 2 CS(P ) such that

�Q0

P Æ ÆP (x) Æ = b 0 Æ �QP . By (C)q, b 0ÆQ(Q) b �10 = ÆQ0(Q0), and hence after restriction,

ÆP (x) Æ conjugates ÆP;P (Q) to ÆP;P (Q0). Since normalizes ÆP;P (Q), this shows that

ÆP (x) conjugates ÆP;P (Q) to ÆP;P (Q0), and hence (since ÆP;P is injective) that xQx�1 =

Q0. We thus have the following commutative diagram:

P ! P

ÆP (x) ! PÆP (x)

�1

! P

Q

�QP#

b 0!Q0

�Q0

P #ÆQ;Q0 (x)

�1

! Q .

�QP#

So if we set b = ÆQ;Q0(x)�1

Æ b 0, then b 2 AutLq(Q) and �QP Æ = b Æ �QP .

The uniqueness of b follows from [5A1, Lemma 3.9]. Finally, for any g 2 Q,b ÆQ(g)b �1 = ÆQ(�(b )(g)) by (C)q, and hence ÆP;P (g) �1 = ÆP;P (�(b )(g)) since mor-

phisms have unique restrictions (Proposition 1.12(b) again). �

Once we have �xed a compatible set of inclusions f�QP g in a linking system Lq, thenfor any ' 2 MorLq(P;Q), and any P 0 � P and Q0 � Q such that �(')(P 0) � Q0, there

is a unique morphism ' 2 MorLq(P0; Q0) such that �QQ0 Æ ' = ' Æ �PP 0. We think of ' as

the restriction of '.

Note, however, that all of this depends on the choice of morphisms �P 2 MorLq(P; S)which satisfy the hypotheses of axiom (D)q, and that not just any lifting of the inclusionincl 2 HomF(P; S) can be chosen. To see why, assume for simplicity that P is alsofully centralized. From the axioms in De�nition 1.9 and Proposition 1.12(b), we seethat if �P ; �

0P 2 MorLq(P; S) are two liftings of incl 2 HomF(P; S), then �

0P = �P Æ ÆP (g)

for some unique g 2 CS(P ). But if �P satis�es the conditions of (D)q, then �0P alsosatis�es those conditions only if g 2 Z(CS(P )).

One situation where the choice of inclusion morphisms is useful is when describingthe fundamental group of jLj or of its p-completion. For any group �, we let B(�)denote the category with one object, and with morphism monoid the group �. Recallthat jLj ' jLqj (Proposition 1.12(a)), so we can work with either of these categories;we will mostly state the results for jLqj. Let the vertex S be the basepoint of jLqj. Foreach morphism ' 2 MorLq(P;Q), let J(') 2 �1(jLqj) denote the homotopy class of theloop �Q�'��P

�1 in jLqj (where paths are composed from right to left). This de�nes afunctor

J : Lq ��! B(�1(jLqj));

where all objects are sent to the unique object of B(�1(jLqj)), and where all inclusionmorphisms are sent to the identity. Let j : S ! �1(jLqj) denote the composite of Jwith the distinguished monomorphism ÆS : S ! AutL(S).

The next proposition describes how J is universal among functors of this type, andalso includes some other technical results for later use about the structure of �1(jLj).


Proposition 1.14. Let (S;F ;L) be a p-local �nite group, and let Lq be the associated

quasicentric linking system. Assume a compatible set of inclusions f�QP g has been chosenfor Lq. Then the following hold.

(a) For any group �, and any functor � : Lq ��! B(�) which sends inclusions tothe identity, there is a unique homomorphism �� : �1(jLqj) ��! � such that � =B(��) Æ J .

(b) For g 2 P � S with P F-quasicentric, J(ÆP (g)) = J(ÆS(g)). In particular,J(ÆP (g)) = 1 in �1(jL

qj) if and only if ÆP (g) is nulhomotopic as a loop basedat the vertex P of jLqj.

(c) If � 2 MorLq(P;Q), and �(�)(x) = y, then j(y) = J(�)j(x)J(�)�1 in �1(jLqj).

(d) If x and y are F-conjugate elements of S, then j(x) and j(y) are conjugate in�1(jLqj).

Proof. Clearly, any functor � : Lq !B(�) induces a homomorphism �� = �1(j�j)between the fundamental groups of their geometric realizations. If � sends inclusionmorphisms to the identity, then � = B(��) Æ J by de�nition of J .

The other points follow easily, using condition (C)q for quasicentric linking systems.Point (d) is shown by �rst reducing to a map between centric subgroups of S whichsends x to y. �

We �nish this introductory section with two unrelated results which will be neededlater in the paper. The �rst is a standard, group theoretic lemma.

Lemma 1.15. Let Q C P be p-groups. If � is a p0-automorphism of P which acts as theidentity on Q and on P=Q, then � = IdP . Equivalently, the group of all automorphismsof P which restrict to the identity on Q and on P=Q is a p-group.

Proof. See [Go, Corollary 5.3.3]. �

The following proposition will only be used in Section 4, but we include it herebecause it seems to be of wider interest. Note, for any fusion system F over S, anysubgroup P � S fully normalized in F , and any P 0 which is S-conjugate to P , that P 0

is also fully normalized in F since NS(P0) is S-conjugate to NS(P ).

Proposition 1.16. Let F be a saturated fusion system over a p-group S. Then forany subgroup P � S, the set of S-conjugacy classes of subgroups F-conjugate to P andfully normalized in F has order prime to p.

Proof. By [BLO2, Proposition 5.5], there is an (S; S)-biset which, when regarded asa set with (S � S)-action, satis�es the following three conditions.

(a) The isotropy subgroup of each point in is of the form

P'def= f(x; '(x)) j x 2 Pg

for some P � S and some ' 2 HomF(P; S).

(b) For each P � S and each ' 2 HomF (P; S), the two structures of (S �P )-set on obtained by restriction and by Id� ' are isomorphic.

(c) jj=jSj � 1 (mod p).


Note that by (a), the actions of S � 1 and 1� S on are both free.

Now �x a subgroup P � S. Set S2 = 1� S for short, and let 0 � be the subsetsuch that 0=S2 = (=S2)

P . In other words, 0 is the set of all x 2 such thatfor each g 2 P , there is some h 2 S satisfying (g; h)�x = x. Since the action of S2on is free, this element h 2 S is uniquely determined for each x 2 0 and g 2 P .Let �(x) : P ! S denote the function such that for each g 2 P , (g; �(x)(g))�x =x. The isotropy subgroup at x of the (P � S)-action is thus the subgroup P�(x) =f(g; �(x)(g)) j g 2 Pg; and by (a), �(x) 2 HomF(P; S). This de�nes a map

� : 0 ! HomF(P; S):

By de�nition, for each ' 2 HomF(P; S), ��1(') is the set of elements of �xed by

P'. By condition (b) above, the action of P � P on induced by the homomorphism1�' 2 Hom(P �P; S�S) is isomorphic to the action de�ned by restriction, and thusj��1(')j = j��1(incl)j. This shows that the point inverses of � all have the same �xedorder k.

Now let RepF(P; S) = HomF(P; S)=Inn(S): the set of S-conjugacy classes of mor-phisms from P to S. Let P be the set of S-conjugacy classes of subgroups F -conjugateto P , and let Pfn � P be the subset of classes of subgroups fully normalized in F . Ifx 2 0 and �(x) = ', then for all s 2 S and g 2 P ,

(1; s)�x = (g; s'(g))�x = (g; cs Æ '(g))�(1; s)�x;

and this shows that �((1; s)�x) = cs Æ �(x). Thus � induces a map

�� : (=S2)P = 0=S2

�=S2��! RepF(P; S)

Im(�)��! P;

where RepF(P; S) = HomF(P; S)=Inn(S). Furthermore, j=S2j � 1 (mod p) by (c),and thus j(=S2)

P j � 1 (mod p).

For each P 0 which is F -conjugate to P , there are jSj=jNS(P0)j distinct subgroups in

the S-conjugacy class [P 0]. Hence there are

jAutF (P )j�jSj=jNS(P0)j

elements of HomF(P; S) whose image lies in [P 0]. Since each of these is the image of kelements in 0, this shows that

j��1([P 0])j = k�jAutF (P )j=jNS(P0)j:

Thus jNS(P0)j��k�jAutF (P )j for all P 0 F -conjugate to P , and so ��1([P 0]) has order a

multiple of p if P 0 is not fully normalized in F (if [P 0] 2 PrPfn). Hence

j��1(Pfn)j � j(=S2)P j � 1 (mod p) :

So if we set m = jNS(P0)j for [P 0] 2 Pfn (i.e., the maximal value of jNS(P

0)j for P 0

F -conjugate to P ), then

j��1(Pfn)j = jPfnj��k�jAutF(P )j

m

�;

and thus jPfnj is prime to p. �

Using a similar argument, one can also show that the set RepfcF(P; S) of elements ofRepF (P; S) whose image is fully centralized also has order prime to p.


2. The fundamental group of jLj^p

The purpose of this section is to give a simple description of the fundamental groupof jLj^p , for any p-local �nite group (S;F ;L), purely in terms of the fusion system F .The result is analogous to the (hyper-) focal subgroup theorem for �nite groups, as weexplain below.

In Section 1, we de�ned a functor J : Lq !B(�1(jLj)), for any p-local �nite group(S;F ;L), and a homomorphism j = J Æ ÆS from S to �1(jLj). Let � : S ! �1(jLj^p ) bethe composite of j with the natural homomorphism from �1(jLj) to �1(jLj^p ).

In [BLO2, Proposition 1.12], we proved that � : S ! �1(jLj^p ) is a surjection. In

this section, we will show that Ker(�) is the hyperfocal subgroup of F , de�ned by Puig[Pu4] (see also [Pu3]).

De�nition 2.1. For any saturated fusion system F over a p-group S, the hyperfocalsubgroup of F is the normal subgroup of S de�ned by

OpF(S) =

g�1�(g)

�� g 2 P � S; � 2 Op(AutF(P ))�:

We will prove, for any p-local �nite group (S;F ;L), that �1(jLj^p )�= S=Op

F(S). Thisis motivated by Puig's hyperfocal theorem, and we will also need that theorem in orderto prove it. Before stating Puig's theorem, we �rst recall the standard focal subgrouptheorem. If G is a �nite group and S 2 Sylp(G), then this theorem says that S\ [G;G](the focal subgroup) is the subgroup generated by all elements of the form x�1y forx; y 2 S which are G-conjugate (cf. [Go, Theorem 7.3.4] or [Suz2, 5.2.8]).

The quotient group S=(S \ [G;G]) is isomorphic to the p-power torsion subgroupof G=[G;G], and can thus be identi�ed as a quotient group of the maximal p-groupquotient G=Op(G). Since G=Op(G) is a p-group, G = S�Op(G), and hence G=Op(G) �=S=(S \Op(G)). Hence S \Op(G) � S \ [G;G]. This subgroup S \Op(G) is what Puigcalls the hyperfocal subgroup, and is described by the hyperfocal subgroup theorem interms of S and fusion.

For S 2 Sylp(G) as above, let OpG(S) be the normal subgroup of S de�ned by

OpG(S) = Op

FS(G)(S) =

g�1�(g)

�� g 2 P � S; � 2 Op(AutG(P ))�

=[g; x]

�� g 2 P � S; x 2 NG(P ) of order prime to p�:

Lemma 2.2 ([Pu3]). Fix a prime p, a �nite group G, and a Sylow subgroup S 2Sylp(G). Then O

pG(S) = S \Op(G).

Proof. This is stated in [Pu3, x1.1], but the proof is only sketched there, and so weelaborate on it here. Following standard notation, for any P � S and any A � Aut(P ),we write [P;A] = hx�1�(x) j x 2 P; � 2 Ai. Thus Op

G(S) is generated by the subgroups[P;Op(AutG(P ))] for all P � S. It is clear that Op

G(S) � S \Op(G); the problem is toprove the opposite inclusion.

Set G� = Op(G) and S� = S \ G� for short. Then [G�; G�] has index prime to pin G�, so it contains S�. By the focal subgroup theorem (cf. [Go, Theorem 7.3.4]),applied to S� 2 Sylp(G�), S� is generated by all elements of the form x�1y for x; y 2 S�which are G�-conjugate. Combined with Alperin's fusion theorem (in Alperin's originalversion [Al] or in the version of Theorem 1.5(a)), this implies that S� is generated byall subgroups [P;NG�(P )] for P � S� such that NS�(P ) 2 Sylp(NG�(P )). (This lastcondition is equivalent to P being fully normalized in FS�(G�).) Also, NG�(P ) is


generated by Op(NG�(P )) and the Sylow subgroup NS�(P ), so [P;NG�(P )] is generatedby [P;Op(NG�(P ))] � Op

G(S) and [P;NS�(P )] � [S�; S�]. Thus S� = hOpG(S); [S�; S�]i.

Since OpG(S) is normal in S (hence also normal in S�), this shows that S�=O

pG(S) is

equal to its commutator subgroup, which for a p-group is possible only if S�=OpG(S) is

trivial, and hence S� = OpG(S). �

By Proposition 1.14, the key to getting information about �1(jLj) is to constructfunctors from L or Lq to B(�), for a group �, which send inclusions to the identity.The next lemma is our main inductive tool for doing this. Whenever � 2 MorLq(P; P

0)and � 2 MorLq(Q;Q

0) are such that P � Q, P 0 � Q0, we write � = �jP to mean that

� is the restriction of � in the sense de�ned in Section 1; i.e., �Q0

P 0 Æ � = � Æ �QP .

Lemma 2.3. Fix a p-local �nite group (S;F ;L), and let Lq be its associated quasi-

centric linking system. Assume a compatible set of inclusions f�QPg has been chosenfor Lq. Let H0 be a set of F-quasicentric subgroups of S which is closed under F-conjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgroupsmaximal among those not in H0, set H = H0 [ P, and let LH0 � LH � Lq be the fullsubcategories with these objects. Assume, for some group �, that

�0 : LH0 ��! B(�)

is a functor which sends inclusions to the identity. Fix some P 2 P which is fullynormalized in F , and �x a homomorphism �P : AutLq(P ) ! �. Assume that

(�) for all P � Q � NS(P ) such that Q is fully normalized in NF (P ), and for all� 2 AutLq(P ) and � 2 AutLq(Q) such that � = �jP , �P (�) = �0(�).

Then there is a unique extension of �0 to a functor � : LH ! B(�) which sends inclu-sions to the identity, and such that �(�) = �P (�) for all � 2 AutF(P ).

Proof. The uniqueness of the extension is an immediate consequence of Theorem 1.5(a)(Alperin's fusion theorem).

To prove the existence of the extension �, we �rst show that (�) implies the following(a prori stronger) statement:

(��) for all Q;Q0 � S which strictly contain P , and for all � 2 MorLq(Q;Q0) and

� 2 AutLq(P ) such that � = �jP , �P (�) = �0(�).

To see this, note �rst that it suÆces to consider the case where P is normal inQ and Q0.By assumption, �(�)(P ) = �(�)(P ) = P , hence �(�)(NQ(P )) � NQ0(P ), and therefore

� restricts to a morphism � 2 MorLq(NQ(P ); NQ0(P )) by Proposition 1.12(b) (applied

with ' = � Æ �QNQ(P ) and = �Q0

NQ0 (P )). Since NQ(P ); NQ0(P ) P , by the induction

hypothesis, �0(�) = �0(�), so we are reduced to proving that �P (�) = �0(�).

Thus � 2 MorNLq (P )(Q;Q0). We now apply Alperin's fusion theorem (Theorem

1.5(a)) to the morphism �(�) in the fusion system NF(P ) (which is saturated by [BLO2,Proposition A.6]). Thus �(�) = 'k Æ � � �Æ'1, where each 'i 2 HomNF (P )(Qi�1; Qi) is therestriction to Qi�1 of an automorphism b'i 2 AutNF (P )(Ri), where Ri � Qi�1; Qi is anNF(P )-centric subgroup of NS(P ) which is fully normalized in NF (P ), and where Q =Q0 and Q

0 = Qk. Each Ri contains P , and hence is F -centric by [BLO2, Lemma 6.2].

For each b'i, also regarded as an automorphism in F , we choose a lifting b�i 2 AutL(Ri),and let �i 2 MorLq(Qi�1; Qi) be its restriction. By (A)q, � = �k Æ � � � Æ �1 Æ ÆQ(g) for


some g 2 CS(Q), and hence

�0(�) = �0(�k) � � ��0(�1) � �0(ÆQ(g)) = �0(b�k) � � ��0(b�1) � �0(ÆNS(P )(g))= �P (b�kjP ) � � ��P (b�1jP ) � �P (IdP ) = �P (�jP ) = �P (�) ;

where the third equality follows from (�). This �nishes the proof of (��).

We can now extend � to be de�ned on all morphisms in LH not in LH0 . Fix sucha morphism ' 2 MorLq(P1; Q). Set P2 = �(')(P1) � Q; then P1; P2 2 P, and' = �QP2 Æ '0 for some unique '0 2 IsoLq(P1; P2). By Lemma 1.3 (and then liftingto the linking category), there are isomorphisms 'i 2 IsoLq(NS(Pi); Ni), for someNi � NS(P ) containing P , which restrict to isomorphisms 'i 2 IsoLq(Pi; P ). Set = '2 Æ'

0Æ'�11 2 AutLq(P ). We have thus decomposed '0 as the composite '�12 Æ Æ'1,

and can now de�ne

�(') = �('0) = �0('2)�1��P ( )��0('1):

Now let '0 = ('02)�1

Æ 0 Æ'01 be another such decomposition, where '0i is the restriction

of '0i 2 IsoLq(NS(Pi); N0i). We thus have a commutative diagram

P '1

P1'01 ! P

P

#

'2P2

'0

#'02 ! P ,

0

#

where for each i, 'i and '0i are restrictions of isomorphisms 'i and '

0i de�ned on NS(Pi).

To see that the two decompositions give the same value of �('), it remains to showthat

�P ( 0)��0('

01 Æ ('1)

�1) = �0('02 Æ ('2)

�1)��P ( ):

And this holds since �0('0i Æ ('i)

�1) = �P ('0i Æ ('i)

�1) by (��).

We have now de�ned � on all morphisms in LH, and it sends inclusion morphismsto the identity by construction. By construction, � sends composites to products, andthus the proof is complete. �

Lemma 2.3 provides the induction step when proving the following proposition, whichis the main result needed to compute �1(jLj^p ).

Proposition 2.4. Fix a p-local �nite group (S;F ;L), and let Lq be its associatedquasicentric linking system. Assume a compatible set of inclusions f�QP g has been chosenfor Lq. Then there is a unique functor

� : Lq ��! B(S=OpF (S))

which sends inclusions to the identity, and such that �(ÆS(g)) = g for all g 2 S.

Proof. The functor � will be constructed inductively, using Lemma 2.3. Let H0 �Ob(Lq) be a subset (possibly empty) which is closed under F -conjugacy and over-groups. Let P be an F -conjugacy class of F -quasicentric subgroups maximal amongthose not in H0, set H = H0 [ P, and let LH0 � LH � Lq be the full subcategorieswith these objects. Assume that

�0 : LH0 ��! B(S=Op

F(S))

has already been constructed, such that �0(ÆS(g)) = g for all g 2 S (if S 2 H0), andsuch that �0 sends inclusions to the identity.


Fix P 2 P which is fully normalized in F , and let ÆP;P : NS(P ) ��! AutLq(P )be the homomorphism of Proposition 1.13. Then Im(ÆP;P ) is a Sylow p-subgroup ofAutLq(P ), since AutS(P ) 2 Sylp(AutF(P )) by axiom (I). We identify NS(P ) as asubgroup of AutLq(P ) to simplify notation. Then

AutLq(P )=Op(AutLq(P )) �= NS(P )

Æ�NS(P ) \O

p(AutLq(P ))�= NS(P )=N0;

where by Lemma 2.2, N0 is the subgroup generated by all commutators [g; x] for g 2Q � NS(P ), and x 2 NAutLq (P )(Q) of order prime to p. In this situation, conjugationby x lies in AutF(Q) by Proposition 1.13(d), and thus [g; x] = g�cx(g)

�1 2 OpF(S). We

conclude that N0 � OpF(S), and hence that the inclusion of NS(P ) into S extends to a

homomorphism

�P : AutLq(P ) ��! S=OpF(S):

We claim that condition (�) in Lemma 2.3 holds for �0 and �P . To see this, �xP � Q � S such that P C Q and Q is fully normalized in NF(P ), and �x � 2 AutLq(P )and � 2 AutLq(Q) such that � = �jP . We must show that �P (�) = �0(�). Uponreplacing � by �k and � by �k for some appropriate k � 1 (mod p), we can assume thatboth automorphisms have order a power of p. Since Q is fully normalized, AutNS(P )(Q)is a Sylow subgroup of AutNF (P )(Q). Hence (since any two Sylow p-subgroups ofa �nite group G are conjugate by an element of Op(G)), there is an automorphism

2 Op(AutNLq (P )(Q)) such that � �1 = ÆQ(g) for some g 2 NS(Q) \ NS(P ). Inparticular, �0( ) = 1 since it is a composite of automorphisms of order prime to p. Set = jP ; then 2 Op(AutLq(P )) and hence �P ( ) = 1. Using axiom (C)q and Lemma

1.12, we see that � �1 = ÆP (g); and thus (since �P ( ) = 1 and �0( ) = 1) that

�0(�) = �0(ÆQ(g)) = g = �P (ÆP (g)) = �P (�):

Thus, by Lemma 2.3, we can extend �0 to a functor de�ned on LH. Upon continuingthis procedure, we obtain a functor � de�ned on all of Lq. �

For any �nite group G, �1(BG^p )�= G=Op(G). (This is implicit in [BK, xVII.3], and

shown explicitly in [BLO1, Proposition A.2].) Hence our main result in this sectioncan be thought of as the hyperfocal theorem for p-local �nite groups.

Theorem 2.5 (Hyperfocal subgroup theorem for p-local �nite groups). Let (S;F ;L)be a p-local �nite group. Then

�1(jLj^p )�= S=Op

F(S):

More precisely, the natural map � : S ! �1(jLj^p ) is surjective, and Ker(�) = OpF(S).

Proof. Let � : L ��! B(S=OpF(S)) be the functor of Proposition 2.4, and let j�j be

the induced map between geometric realizations. Since jB(S=OpF(S))j is the classifying

space of a �nite p-group and hence p-complete, j�j factors through the p-completionjLj^p . Consider the following commutative diagram:

Sj! �1(jLj)

HHHHHHHH

�j

HHHHHHHH

�1(j�j)

j

�1(jLj^p )

(�)^p #��=

�1(j�j^p )! S=Op

F(S) .


Here, � is surjective by [BLO2, Proposition 1.12]. Also, by construction, the composite�� Æ � = �1(j�j) Æ j is the natural projection. Thus Ker(�) � Op

F(S), and it remains toshow the opposite inclusion.

Fix g 2 P � S and � 2 AutF(P ) such that � has order prime to p; we want toshow that g�1�(g) 2 Ker(�). Since Ker(�) is closed under F -conjugacy (Proposition1.14(d)), for any ' 2 IsoF(P; P

0), g�1�(g) is in Ker(�) if '(g�1�(g)) is in Ker(�). Inparticular, since '(g�1�(g)) = '(g)�1�('�'�1)('(g)), we can assume that P is fullycentralized in F . Then, upon extending � to an automorphism of P �CS(P ), whichcan be assumed also to have order prime to p (replace it by an appropriate power ifnecessary), we can assume that P is F -centric. In this case, by Proposition 1.14(c),j(g) and j(�(g)) are conjugate in �1(jLj) by an element of order prime to p, and henceare equal in �1(jLj^p ) since this is a p-group. This shows that g

�1�(g) 2 Ker(�), and�nishes the proof of the theorem. �

The following result, which will be useful in Section 5, is of a similar nature, butmuch more elementary.

Proposition 2.6. Fix a p-local �nite group (S;F ;L), and let Lq be its associatedquasicentric linking system. Then the induced maps

�1(jLj) ��! �1(jFcj) and �1(jL

qj) ��! �1(jFqj)

are surjective, and their kernels are generated by elements of p-power order.

Proof. We prove this for jLqj and jF qj; a similar argument applies to jLj and jF cj.Recall from Section 1 that we can regard �1(jLqj) as the group generated by the loopsJ(�), for � 2 Mor(Lq), with relations given by composition of morphisms and makinginclusion morphisms equal to 1. In a similar way, we regard �1(jF

qj) as being generatedby loops J(') for ' 2 Mor(F q). Since every morphism � 2 HomFq(P;Q) has a liftingto a morphism of Lq, the map �# : �1(jLqj) ! �1(jF qj) induced by the projectionfunctor � : Lq !F q is an epimorphism.

Write JC for the normal subgroup of �1(jLqj) generated by the loops j(g) = J(ÆP (g)),for all F -quasicentric subgroups P � S and all g 2 CS(P ). In particular, JC isgenerated by elements of p-power order. Since �#(j(g)) = 1, JC is contained in thekernel of �#, and we have a factorization b�# : �1(jLqj)=JC ! �1(jF qj).

De�ne an inverse s : �1(jF qj) ! �1(jLqj)=JC as follows. Given a morphism � 2HomFq(P;Q), choose a lifting �� in Lq, and set s(�) = [J(��)]. If ��; ��0 2 HomLq(P;Q)are two liftings of �, then there is an isomorphism � : P 0 ! P in Lq where P 0 is fullycentralized, and an element g 2 CS(P

0), such that ��0 = �� Æ (� Æ ÆP 0(g) Æ ��1). Since

�ÆÆP 0(g)Æ��1 represents a loop that belongs to JC, [J(��)] = [J(��0)]. Thus the de�nition

of s does not depend on the choice of ��. It remains to show that s preserves the relationsamong the generators. But we clearly have that s([J(inclQP )]) = [J(�QP )] = 1. Also, if �and � are composable morphisms of F q, and �� and �� are liftings to Lq, then �� Æ �� is alifting of � Æ �, and hence s(�)s(�) = [J(��)][J( ��)] = [J(�� Æ ��)] = s(� Æ �).

This shows that �# induces an isomorphim �1(jLqj)=JC �= �1(jF qj). �

3. Subsystems with p-solvable quotient

In this section, we prove some general results about subsystems of saturated fusionsystems and p-local �nite subgroups: subsystems with p-group quotient or quotient of


order prime to p. These will then be used in the next two sections to prove some morespeci�c theorems.

It will be convenient in this section to write \p0-group" for a �nite group of orderprime to p. Recall that a p-solvable group is a group G with normal series 1 = H0 C

H1 C � � � C Hk = G such that each Hi=Hi�1 is a p-group or a p0-group. As oneconsequence of the results of this section, we show that for any p-local �nite group(S;F ;L), and any homomorphism � from �1(jLj) to a �nite p-solvable group, there isanother p-local �nite group (S0;F0;L0) such that jL0j is homotopy equivalent to thecovering space of jLj with fundamental group Ker(�).

We start with some de�nitions. Recall that for any �nite group G, Op0(G) andOp(G) are the smallest normal subgroups of G of index prime to p and of p-powerindex, respectively. Equivalently, Op0(G) is the subgroup generated by elements of p-power order in G, and Op(G) is the subgroup generated by elements of order prime top in G.

We want to identify the fusion subsystems of a given fusion system which are anal-ogous to subgroups of G which contain Op0(G) or Op(G). This motivates the followingde�nitions.

De�nition 3.1. Let F be a saturated fusion system over a p-group S. Let (S 0;F 0) �(S;F) be a saturated fusion subsystem; i.e., S 0 � S is a subgroup, and F 0 � F is asubcategory which is a saturated fusion system over S 0.

(a) (S 0;F 0) is of p-power index in (S;F) if S 0 � OpF(S), and AutF 0(P ) � Op(AutF(P ))

for all P � S 0. Equivalently, a saturated fusion subsystem F 0 � F over S 0 �OpF(S) has p-power index if it contains all F-automorphisms of order prime to p

of subgroups of S 0.

(b) (S 0;F 0) is of index prime to p in (S;F) if S 0 = S, and AutF 0(P ) � Op0(AutF(P ))for all P � S. Equivalently, a saturated fusion subsystem F 0 � F over S has indexprime to p if it contains all F-automorphisms of p-power order.

This terminology has been chosen for simplicity. Subsystems \of p-power index" or\of index prime to p" are really analogous to subgroups H � G which contain normalsubgroups of G of p-power index or index prime to p, respectively. For example, if F isa saturated fusion system over S, and F 0 � F is the fusion system of S itself (i.e., theminimal fusion system over S), then F 0 is not in general a subsystem of index primeto p under the above de�nition, despite the inclusion F 0 � F being analogous to theinclusion of a Sylow p-subgroup in a group.

Over the next three sections, we will classify all saturated fusion subsystems of p-power index, or of index prime to p, in a given saturated fusion system. In bothcases, there will be a minimal such subsystem, denoted Op(F) or Op0(F); and thesaturated fusion subsystems of the given type will be in bijective correspondence withthe subgroups of a given p-group or p0-group.

The following terminology will be useful for describing some of the categories wehave to work with.

De�nition 3.2. Fix a �nite p-group S.


(a) A restrictive category over S is a category F such that Ob(F) is the set of subgroupsof S, such that all morphisms in F are group monomorphisms between the sub-groups, and with the following additional property: for each P 0 � P � S and Q0 �Q � S, and each ' 2 HomF(P;Q) such that '(P 0) � Q0, 'jP 0 2 HomF(P

0; Q0).

(b) A restrictive category F over S is normalized by an automorphism � 2 AutF(S)if for each P;Q � S, and each ' 2 HomF(P;Q), �'�

�1 2 HomF(�(P ); �(Q)).

(c) For any restrictive category F over S and any subgroup A � Aut(S), hF ; Ai is thesmallest restrictive category over S which contains F together with all automor-phisms in A and their restrictions.

By de�nition, any restrictive category is required to contain all inclusion homomor-phisms (restrictions of IdS). The main di�erence between a restrictive category over Sand a fusion system over S is that the restrictive subcategory need not contain FS(S).

When F is a fusion system over S, then an automorphism � 2 Aut(S) normalizesF if and only if it is fusion preserving in the sense used in [BLO1]. For the purposesof this paper, it will be convenient to use both terms, in di�erent situations.

We next de�ne the following subcategories of a given fusion system F .

De�nition 3.3. Let F be any fusion system over a p-group S.

(a) Op�(F) � F denotes the smallest restrictive subcategory of F whose morphism set

contains Op(AutF(P )) for all subgroups P � S.

(b) Op0

� (F) � F denotes the smallest restrictive subcategory of F whose morphism setcontains Op0(AutF(P )) for all subgroups P � S.

By de�nition, for subgroups P;Q, a morphism ' 2 HomF (P;Q) lies in Op0

� (F) if andonly if it is a composite of morphisms which are restrictions of elements ofOp0(AutF(R))for subgroups R � S. Morphisms in Op

�(F) are described in a similar way.

The subcategory Op�(F) is not, in general, a fusion system | and this is why we

had to de�ne restrictive categories. The subcategory Op0

� (F) is always a fusion system(since AutS(P ) � Op0(AutF(P )) for all P � S), but it is not, in general, saturated.The subscripts \�" have been put in as a reminder of these facts, and as a contrastwith the notation Op(F) and Op0(F) which will be used to denote certain minimalsaturated fusion systems.

We now check some of the basic properties of the subcategories Op�(F) and O

p0

� (F),stated in terms of these de�nitions.

Lemma 3.4. The following hold for any fusion system F over a p-group S.

(a) Op�(F) and O

p0

� (F) are normalized by AutF(S).

(b) If F is saturated, then F = hOp�(F);AutF (S)i = hO

p0

� (F);AutF(S)i.

(c) If F 0 � F is any restrictive subcategory normalized by AutF(S) and such thatF = hF 0;AutF(S)i, then for each P;Q � S and ' 2 HomF(P;Q), there aremorphisms � 2 AutF(S), '

0 2 HomF 0(�(P ); Q), and '00 2 HomF 0(P; �

�1Q), suchthat ' = '0 Æ �jP = � Æ '00.

Proof. To simplify notation in the following proofs, for � 2 AutF(S), we write � incomposites of morphisms between subgroups of S, rather than specifying the appro-priate restriction each time.


(a) Set F 0 = Op�(F) or O

p0

� (F). Let � 2 AutF(S) and let ' 2 HomF 0(P;Q). We mustshow that �'��1 2 HomF 0(�(P ); �(Q)). By de�nition of F 0, there are subgroups

P = P0; P1; : : : ; Pk = '(P ) � Q;

subgroups P 1; : : : ; P k, and automorphisms �i 2 Op(AutF(P i)) (if F 0 = Op�(F)) or

�i 2 Op0(AutF(P i)) (if F 0 = Op0

� (F)), such that Pi�1; Pi � P i, �i(Pi�1) = Pi, and

' = inclQPk Æ(�kjPk�1) Æ � � � Æ (�2jP1) Æ (�1jP0):

Let P 0i = �(Pi), P

0i = �(P i), and �0i = ��i�

�1 2 Op(AutF(P0i)) or O

p0(AutF(P0i)).

Then�'��1 = incl

�(Q)P 0k

Æ(�0kjP 0k�1) Æ � � � Æ (�02jP 01) Æ (�

01jP 00)

is in HomF 0(�(P ); �(Q)), as required.

(b1) Now assume that F is saturated. We show here that F = hOp0

� (F);AutF(S)i;i.e., that each ' 2 Mor(F) is a composite of morphisms in Op0

� (F) and in AutF(S). ByAlperin's fusion theorem for saturated fusion systems (Theorem 1.5(a)), it suÆces toprove this when ' 2 AutF (P ) for some P � S which is F -centric and fully normalized.The result clearly holds if P = S. So we can assume that P � S, and also assumeinductively that the lemma holds for every automorphism of any subgroup P 0 � S suchthat jP 0j > jP j.

Consider the subgroup

K = 'AutS(P )'�1 = f'cg'

�1 2 AutF(P ) j g 2 NS(P )g:

Then K is a p-subgroup of AutF(P ), and hence K � AutOp

0

� (F)(P ). Since P is fully

normalized in F , AutS(P ) 2 Sylp(AutF (P )). Thus K and AutS(P ) are both Sylowp-subgroups of Aut

Op0

� (F)(P ), and there is some � 2 Aut

Op0

� (F)(P ) such that �K��1 �

AutS(P ). In other words,

N�'def= fg 2 NS(P ) j (�')cg(�')

�1 2 AutS(P )g = NS(P ):

So by condition (II) in De�nition 1.2, � Æ ' can be extended to a homomorphism 2HomF (NS(P ); S). By the induction hypothesis (and since NS(P ) P ), , and hence� Æ ', are composites of morphisms in NF(S) and in Op0

� (F). Also, � 2 AutOp

0

� (F)(P )

by assumption, and hence ' is a composite of morphisms in these two subcategories.

(b2) To see that F = hOp�(F);AutF(S)i when F is saturated, it again suÆces to

restrict to the case where ' 2 AutF(P ) for some P � S which is F -centric andfully normalized. But in this case, AutF(P ) is generated by its Sylow p-subgroupAutS(P ) � AutNF (S)(P ), together with O

p(AutF(P )) � AutOp�(F)(P ).

(c) Since F = hF 0;AutF (S)i, every morphism in F is a composite of morphisms inF 0 and restrictions of automorphisms in AutF(S). Assume ' 2 HomF (P;Q) is thecomposite

P = P0�0��! Q0

'1��! P1�1��! Q1 ��! : : : ��! Pn

�n��! Qn = Q;

where �i 2 AutF(S), �i(Pi) = Qi, and 'i 2 HomF 0(Qi�1; Pi). Write �j;i = �j � � ��i forany i � j, and set � = �n;0 = �n � � ��0. Then

' = � Æ (�n�1;0�1'n�n�1;0) � � � (�0;0

�1'1�0;0) = (�n;n'n�n;n�1) � � � (�n;1'1�n;1

�1) Æ �;

where each �n;i'i�n;i�1 and each �i;0

�1'i+1�i;0 is a morphism in F 0 since F 0 is normal-ized by AutF (S). �


The following lemma will also be needed later.

Lemma 3.5. Let F be a saturated fusion system over a p-group S. Fix a normalsubgroup S0 C S which is strongly F-closed; i.e., no element of S0 is F-conjugate toany element of SrS0. Let (S0;F0) be a saturated fusion subsystem of (S;F). Then forany P � S which is F-centric and F-radical, P \ S0 is F0-centric.

Proof. Assume P � S is F -centric and F -radical, and set P0 = P\S0 for short. Choosea subgroup P 0

0 � S0 which is F -conjugate to P0 and fully normalized inF . In particular,by (I), P 0

0 is fully centralized in F . By Lemma 1.3, there is ' 2 HomF(NS(P0); NS(P00))

such that '(P0) = P 00. Set P 0 = '(P ); thus P 0 is F -conjugate to P (so P 0 is also

F -centric and F -radical), and P 00 = P 0 \ S0 since S0 is strongly closed. For any

P 000 � S0 which is F0-conjugate to P0 (hence F -conjugate to P

00), there is a morphism

2 HomF(P000 �CS(P

000 ); P

00�CS(P

00)) such that (P 00

0 ) = P 00 (by axiom (II)), and then

(CS0(P000 )) � CS0(P

00). So if CS0(P

00) = Z(P 0

0), then CS0(P000 ) = Z(P 00

0 ) for all P000 which

is F0-conjugate to P0, and P0 is F0-centric.

We are thus reduced to showing that CS0(P00) = Z(P 0

0); and without loss of generality,we can assume that P 0 = P and P 0

0 = P0. Since S0 is strongly closed, every � 2AutF (P ) leaves P0 invariant. Let Aut0F(P ) � AutF(P ) be the subgroup of elementswhich induce the identity on P0 and on P=P0. This is a normal subgroup of AutF(P )since all elements of AutF(P ) leave P0 invariant, and is also a p-subgroup by Lemma1.15. Thus Aut0F(P ) � Op(AutF(P )), and hence Aut0F(P ) � Inn(P ) since P is F -radical.

We want to show that CS0(P0) = Z(P0). Fix any x 2 CS0(P0), and assume �rstthat the coset x�Z(P0) 2 CS0(P0)=Z(P0) is �xed by the conjugation action of P . Thusx 2 S0, [x; P0] = 1, and [x; P ] � Z(P0), so cx 2 Aut0F(P ) � Inn(P ), and xg 2 CS(P ) forsome g 2 P . Since P is F -centric, this implies that xg 2 P , so x 2 CS0(P0)\P = Z(P0).In other words, [CS0(P0)=Z(P0)]

P = 1, so CS0(P0)=Z(P0) = 1, and thus CS0(P0) =Z(P0). �

The motivation for the next de�nition comes from considering the situation whicharises when one is given a saturated fusion system F with an associated quasicentriclinking system Lq, and a functor Lq !B(�) which sends inclusions to the identity (orequivalently a homomorphism from �1(jLqj) to �) for some group �. Such a functor

is equivalent to a function �: Mor(Lq) ! � which sends composites to products

and sends inclusions to the identity; and for any H � �, ��1(H) = Mor(LqH) forsome subcategory LqH � L

q with the same objects. Let F qH � Fq be the image of LqH

under the canonical projection; then in some sense (to be made precise later), F qH is asubsystem of F q of index [�:H].

What we now need is to make sense of such \inverse image subcategories" of thefusion system F , when we are not assuming that we have an associated linking system.

LetSub(�) denote the set of nonempty subsets of �. Given a function � as above, thereis an obvious associated function �: Mor(F q) !Sub(�), which sends a morphism

� 2 Mor(F q) to �(��1(�)). Here, � denotes the natural projection from Lq to F q.

Moreover, � also induces a homomorphism � = � Æ ÆS from S to �. The maps �and � are closely related to each other, and satisfy certain properties, none of whichdepend on the existence (or choice) of a quasicentric linking system associated to F .In fact, we will see that the data encoded in such a pair of functions, if they satisfy theappropriate conditions, suÆces to describe precisely what is meant by \inverse image


subcategories" of fusion systems; and to show that under certain restrictions, thesecategories are (or generate) saturated fusion subsystems.

De�nition 3.6. Let F be a saturated fusion system over a p-group S, and let F0 �F q be any full subcategory such that Ob(F0) is closed under F-conjugacy. A fusionmapping triple for F0 consists of a triple (�; �;�), where � is a group, � : S ��! � isa homomorphism, and

�: Mor(F0) ��! Sub(�);

is a map which satis�es the following conditions for all subgroups P;Q;R � S whichlie in F0:

(i) For all P'��! Q

��! R in F0, and all x 2 �( ), �( ') = x��(').

(ii) If P is fully centralized in F , then �(IdP ) = �(CS(P )).

(iii) If ' = cg 2 HomF(P;Q), where g 2 NS(P;Q), then �(g) 2 �(').

(iv) For all ' 2 HomF (P;Q), all x 2 �('), and all g 2 P , x�(g)x�1 = �('(g)).

For any fusion mapping triple (�; �;�) and any H � �, we let F�H � F be the smallestrestrictive subcategory which contains all ' 2 Mor(F q) such that �(') \ H 6= ;. LetFH � F�H be the full subcategory whose objects are the subgroups of ��1(H).

When � is the trivial homomorphism (which is always the case when � is a p0-group),then a fusion mapping triple (�; �;�) on a subcategory F0 � F

q is equivalent to afunctor from F0 to B(�); i.e., �(') contains just one element for all ' 2 Mor(F0). By(i), it suÆces to show this for identity morphisms; and by (ii), j�(IdP )j = 1 if P is fullycentralized. For arbitrary P , it then follows from (i), together with the assumption(included in the de�nition of Sub(�)) that �(') 6= ; for all '.

The following additional properties of fusion mapping triples will be needed.

Lemma 3.7. Fix a saturated fusion system F over a p-group S, let F0 be a fullsubcategory such that Ob(F0) is closed under F-conjugacy, and let (�;�; �) be a fusionmapping triple for F0. Then the following hold for all P;Q;R 2 Ob(F0).

(v) �(IdP ) is a subgroup of �, and � restricts to a homomorphism

�P : AutF(P ) !N�(�(IdP ))=�(IdP ):

Thus �P (�) = �(�) (as a coset of �(IdP )) for all � 2 AutF(P ).

(vi) For all P'��! Q

��! R in F0, and all x 2 �('), �( ') � �( )�x, with equality

if '(P ) = Q. In particular, if P � Q, then �( jP ) � �( ).

(vii) Assume S 2 Ob(F0). Then for any ' 2 HomF(P;Q), any � 2 AutF(S), and anyx 2 �(�), �(�'��1) = x�(')x�1, where �'��1 2 HomF (�(P ); �(Q)).

Proof. (v) By (i), for any �; � 2 AutF(P ) and any x 2 �(�), �(��) = x��(�). Whenapplied with � = � = IdP , this shows that �(IdP ) is a subgroup of �. (Note here that�(IdP ) 6= ; by de�nition of Sub(�).) When applied with � = ��1, this shows thatx�1 2 �(��1) if x 2 �(�). Hence �(�) = x��(IdP ) implies that �(��1) = �(IdP )�x�1

and �(��1) = x�1��(IdP ), and thus that each �(�) is a right coset as well as a leftcoset. Thus �(�) � N�(�(IdP )) for all � 2 AutF (P ), and the induced map �P isclearly a homomorphism.


(vi) By (i), �( ') � �( )��(') for any pair of composable morphisms '; in F0. Inparticular, �( ') � �( )�x if x 2 �('). If ' is an isomorphism, then 1 2 �(IdP ) =x��('�1) by (v) and (i), so x�1 2 �('�1). This gives the inclusions

�( ) = �( ''�1) � �( ')�x�1 � �( )xx�1;

and hence these are both equalities. The last statement is the special case where P � Qand ' = inclQP ; 1 2 �(inclQP ) by (iii).

(vii) For x 2 �(�), �(�') = x��(') = �(�'��1)�x by (i) and (vi). �

We are now ready to prove the main result about fusion mapping triples.

Proposition 3.8. Let F be a saturated fusion system over a �nite p-group S. Let(�; �;�) be any fusion mapping triple on F q, where � is a p-group or a p0-group, and

� : S ��! � and �: Mor(F q) ��! Sub(�);

Then the following hold for any subgroup H � �, where we set SH = ��1(H).

(a) FH is a saturated fusion system over SH.

(b) If � is a p-group, then a subgroup P � SH is FH-quasicentric if and only if it isF-quasicentric. Also, F�1 � Op

�(F).

(c) If � is a p0-group, then SH = S. A subgroup P � S is FH-centric (fully centralizedin FH, fully normalized in FH) if and only if it is F-centric (fully centralized inF , fully normalized in F). Also, F�1 � Op0

� (F).

Proof. Throughout the proof, (i){(vii) refer to the conditions in De�nition 3.6 andLemma 3.7. If X is any set of morphisms of F , then we let �(X) be the union of thesets �(�) for � 2 X. Condition (v) implies that for any P � S and any subgroupA � AutF (P ), �(A) is a subgroup of �.

We �rst prove the following two additional properties of these subcategories:

(1) For each pair of subgroups P;Q � S, and each ' 2 HomF(P;Q), there are Q0 � S,

'0 2 HomF�1(P;Q0), and � 2 AutF(S) such that �(Q0) = Q and ' = (�jQ0) Æ '0.

(2) For all P � S there exists P 0 � S which is fully normalized in F , and ' 2HomF�

1(NS(P ); NS(P

0)) such that '(P ) = P 0.

By (vii), for any � 2 AutF (S), any x 2 �(�), and any ' 2 HomF(P;Q), �(�'��1) =

x�(')x�1. In particular, �(�'��1) = 1 if and only if �(') = 1, and thus F�1 isnormalized by AutF (S).

Let Op0

� (F)c � Op0

� (F) and Op�(F)

c � Op�(F) be the full subcategories whose objects

are the F -centric subgroups of S. By (v), for each F -quasicentric subgroup P � S,AutF�

1(P ) = Ker(�P ) where �P is a homomorphism to a subquotient of �. Hence

AutF�1(P ) contains Op0(AutF(P )) if � is a p0-group or Op(AutF (P )) if � is a p-group.

This shows that F�1 contains either Op0

� (F)c (if � is a p0-group) or Op

�(F)c (if � is a p-

group). Thus F c � hF�1 ;AutF(S)i by Lemma 3.4(b), and so F = hF�1 ;AutF(S)i sinceF is the smallest restrictive category over S which contains F c by Theorem 1.5(a)(Alperin's fusion theorem). Point (1) now follows from Lemma 3.4(c).

To see (2), recall that by Lemma 1.3, if P 00 is any subgroup F -conjugate to Pand fully normalized in F , then there is a morphism '0 2 HomF(NS(P ); NS(P

00))such that '0(P ) = P 00. By (1), '0 = � Æ ' for some � 2 AutF(S) and some ' 2


HomF�1(NS(P ); NS(�

�1(P 00))). Also, the subgroup P 0 = ��1(P 00) is fully normalized inF since P 00 is.

(b) Assume � is a p-group, and �x H � �. Consider the following set of subgroupsof SH :

QH =�P � SH

�� 8P 0 FH-conjugate to P , 8P0 � Q � P 0�CSH (P

0);

8� 2 AutFH (Q) such that �jP 0 = Id, j�j is a power of p:

This set clearly contains all FH-centric subgroups of SH . By Lemma 1.6, if FH issaturated, then QH is precisely the set of FH-quasicentric subgroups. We prove (b)here with \FH-quasicentric" replaced by \element of QH". This is all that will beneeded for the proof that FH is saturated; and once that is shown then (b) (in itsoriginal form) will follow.

Assume P � SH and P =2 QH . Fix P 0 which is FH-conjugate to P , Q � P 0�CSH (P0)

which contains P 0, and IdQ 6= � 2 AutFH (Q) of order prime to p, such that �jP 0 = IdP 0.Then P 0, and hence P , is not F -quasicentric by Lemma 1.6(a).

We have shown that if P is F -quasicentric, then P 2 QH , and it remains to check theconverse. Assume P is not F -quasicentric: �x P 0 � Q � P 0�CS(P

0) and � 2 AutF(Q)as in Lemma 1.6(b). In particular, Q is F -centric, and hence F -quasicentric. SetQ1 = Q \ S1 (where S1 = ��1(1)). Since 1 2 �(�) by (v) (and since j�j is prime top), �(g) = �(�(g)) for all g 2 Q by (iv), and thus �(Q1) = Q1 and (since g�1�(g) 2Ker(�) = S1 for g 2 Q) � induces the identity on Q=Q1. Since j�j is not a power of p,it cannot be the identity on both Q1 and Q=Q1 (Lemma 1.15), and hence �jQ1

6= IdQ1.

Thus P 0 � P 0Q1 � P 0�CSH (P0), �jP 0Q1

is a nontrivial automorphism of P 0Q1 of orderprime to p whose restriction to P 0 is the identity.

Finally, by (1), P is FH-conjugate to a subgroup P 00 for which there is some ' 2IsoF(P

0; P 00) which is the restriction of some ' 2 AutF(S). Since P � SH and is FH-conjugate to P 00, P 00 � SH by (iv) (applied with � 2 IsoFH (P; P

00) and x 2 �(�) \H).Set Q00 = '(P 0Q1) = P 00�'(Q1) � SH and �00 = '�'�1 2 AutF(Q

00). Since � isa p-group and j�00j is prime to p, (v) implies that �(�00) = �(IdQ), and hence that1 2 �(�00) and hence �00 2 AutFH (Q

00). But then P 00 =2 QH , and hence P =2 QH .

It remains to show that F�1 � Op�(F); i.e., to show that

AutF�1(P ) � Op(AutF(P )) (3)

for each P � S. If P � S is F -quasicentric, then (3) holds by (v): AutF�1(P ) is the

kernel of a homomorphism from AutF(P ) to a p-group. If P is not F -quasicentricbut is fully centralized in F , then every automorphism � 2 AutF(P ) of order primeto p extends to an automorphism � 2 AutF(P �CS(P )), which (after replacing it byan appropriate power) can also be assumed to have order prime to p, and hence inOp(AutF(P �CS(P ))). Thus (3) holds for P , since it holds for the F -centric subgroupP �CS(P ). Finally, by (1), every subgroup of S is F�1 -conjugate to a subgroup which isfully centralized in F , and thus (3) holds for all P � S.

(c) This point holds, in fact, without assuming that the fusion system F be saturated.Assume � is a p0-group; then � is the trivial homomorphism, and so SH = ��1(H) = Sfor all H. Fix H � �, and let P be any subgroup of S. Since each FH-conjugacyclass is contained in some F -conjugacy class, any subgroup which is fully centralized(fully normalized) in F is also fully centralized (fully normalized) in FH . By the samereasoning, any F -centric subgroup P � S is also FH-centric.


Conversely, assume P is not fully centralized in F , and let P 0 be a subgroup in the F -conjugacy class of P such that jCS(P 0)j > jCS(P )j. Fix some ' 2 IsoF(P; P

0). By (1),there are a subgroup P 00 � S and isomorphisms � 2 AutF(S) and '

0 2 IsoF1(P; P00),

such that ��1(P 00) = P 0 and ' = (�jP 00) Æ '0. Thus P 00 is FH-conjugate to P , and is

F -conjugate to P 0 via a restriction of an F -automorphism of S. Hence jCS(P 00)j =jCS(P

0)j > jCS(P )j, and this shows that P is not fully centralized in FH . A similarargument shows that if P is not fully normalized in F (or not F -centric), then it is notfully normalized in FH (or not FH-centric).

The proof that F�1 � Op0

� (F) is identical to the proof of the corresponding result in(b).

(a) Fix H � �. Clearly, FH is a fusion system over SH ; we must prove it is saturated.By (b) or (c), each FH-centric subgroup of SH is F -quasicentric. Hence by Theorem1.5(b), it suÆces to prove conditions (I) and (II) in De�nition 1.2 for F -quasicentricsubgroups P � SH . Thus we will be working only with subgroups P � SH for which� is de�ned on HomF(P; S).

Proof of (I): Assume � is a p0-group; thus SH = S. If P is fully normalized in FH,then by (c), P is fully normalized in F , hence it is fully centralized in F and in FH.Also, AutS(P ) 2 Sylp(AutF (P )), and hence AutS(P ) is also a Sylow p-subgroup ofAutFH (P ). This proves (I) in this case.

Now assume � is a p-group. Fix P � SH which is F -quasicentric and fully normalizedin FH , and let � 2 HomF�

1(NS(P ); NS(P

0)) be as in (2). In particular, 1 2 �(�) byde�nition of F�1 ; so by (iv), �(�(NSH (P ))) = �(NSH (P )) � H. Hence �(NSH (P )) �SH \NS(P

0) = NSH (P0), and this is an equality since P is fully normalized in FH . In

particular, this shows that P 0 � SH . The conclusion of (I) holds for P (i.e., P is fullycentralized in FH and AutSH (P ) 2 Sylp(AutFH (P ))) if the conclusion of (I) holds forP 0. So we can assume that P = P 0 is fully normalized in F and in FH .

Fix Q � SH which is FH-conjugate to P and fully centralized in FH, and choosean isomorphism 2 IsoFH (P;Q). After applying (2) again, we can assume that Q isalso fully normalized in F , and hence also fully centralized. Hence extends to some 1 2 HomF(P �CS(P ); Q�CS(Q)), which is an isomorphism since P and Q are both fullycentralized in F . Fix h 2 �( �1) \H and g 2 �( 1). Then gh 2 �(IdQ) = �(CS(Q))by (i) and (ii). Let a 2 CS(Q) be such that �(a) = gh, and set

2 = c�1a Æ 1 2 IsoF(P �CS(P ); Q�CS(Q)):

Thus 2jP = 1jP = since a 2 CS(Q), and h�1 = �(a)�1g 2 �( 2) by (iii) and (i).

By (iv), for all g 2 P �CS(P ), �( 2(g)) = h�1�(g)h, and hence 2(g) 2 H if and only ifg 2 H. Thus 2 sends CSH (P ) onto CSH (Q). Since Q is fully centralized in FH , so isP .

It remains to show that AutSH(P ) 2 Sylp(AutFH (P )). Set P� = �(IdP ) = �(CS(P ))for short (see (ii)). By (v), � restricts to a homomorphism

�P : AutF(P ) ��! N�(P�)=P�:

Set bH =�hP�

�� h 2 H \N�(P�)� N�(P�)=P�;

then AutFH (P ) = �P�1( bH) by de�nition of FH . For any ' = ca 2 AutS(P )\AutFH (P )

(where a 2 NS(P )), �(') = �(a)��(CS(P )) by (ii) and (iii); and thus �(') containssome h 2 H where h = �(b) for some b 2 a�CS(P ). Hence b 2 SH , and ' = cb 2


AutSH (P ). This shows that

AutSH (P ) = AutS(P ) \ AutFH (P ) = AutS(P ) \��1P ( bH):

Also, �P (AutS(P )) = �P (AutF (P )), since AutS(P ) 2 Sylp(AutF (P )) and � is a p-group, and hence

[AutS(P ) : AutSH (P )] = [Im(�P ) : bH] = [AutF(P ) : AutFH (P )]:

Since AutS(P ) 2 Sylp(AutF (P )) (i.e., [AutF(P ) : AutS(P )] is prime to p), this impliesthat AutSH (P ) 2 Sylp(AutFH (P )).

Proof of (II): Fix a morphism ' 2 IsoFH (P;Q), for some F -quasicentric P;Q � SHsuch that Q = '(P ) is fully centralized in FH , and set N' = fg 2 NSH (P ) j'cg'

�1 2AutSH (Q)g. By (2), there is a subgroup Q0 fully normalized in F , and a morphism 2 HomF�

1(NS(Q); NS(Q

0)), such that (Q) = Q0. By condition (II) for the saturatedfusion system F , there is '1 2 HomF(N'; NS(Q

0)) such that '1jP = Æ '. Fix some

x 2 �('1) � �( ') = �(')

(where the last equality holds since 1 2 �( )). Since ' 2 Mor(FH), there is h 2 Hsuch that h 2 �('), and thus

hx�1 2 �(IdQ0) = �(CS(Q0)):

Fix a 2 CS(Q0) such that hx�1 = �(a), and set

'2 = ca Æ '1 2 HomF(N'; NS(Q0)):

Then by (i), h 2 �('2), so '2 2 HomFH (N'; NSH (Q0)); and '2jP = '1jP since

a 2 CS(Q0). Since Q is fully centralized in FH , sends CSH (Q) isomorphically onto

CSH (Q0); and hence (by de�nition of N') '2(N') � Im( ). So '

def= �1 Æ '2 sends N'

into NSH (Q) and extends '. �

We next extend Proposition 3.8 to a result about linking systems and p-local �nitegroups. The main point of the following theorem is that for any p-local �nite group

(S;F ;L) and any epimorphism �1(jLj)b�� , where � is a �nite p-group or p0-group,

there is another p-local �nite group (SH ;FH ;LH) for each subgroup H � �, such thatjLH j is homotopy equivalent to the covering space of jLj whose fundamental group isb��1(H).

Recall that for any p-local �nite group (S;F ;L) with associated quasicentric link-ing system Lq, j : S ! �1(jLj) �= �1(jLqj) denotes the homomorphism induced bythe distinguished monomorphism ÆS : S ! AutL(S), and J : Lq !B(�1(jL

qj))

is the functor which sends morphisms to loops as de�ned in Section 1. Let b� be ahomomorphism from �1(jL

qj) to a group �, and set

� = b� Æ j 2 Hom(S;�) and b� = B(b�) Æ J : Lq ��! B(�):Note that these depend on a choice of a compatible set of inclusions f�QPg for L

q (sinceJ depends on such a choice). For any subgroup H � �, let L�H � L

q be the subcategory

with the same objects and with morphism set b��1(H); and let LqH � L�H be the full

subcategory obtained by restricting to subgroups of SHdef= ��1(H). Finally, let FH be

the fusion system over SH generated by �(LqH) � Fq and restrictions of morphisms,

and let LH � LqH be the full subcategory on those objects which are FH-centric.


Theorem 3.9. Let (S;F ;L) be a p-local �nite group, let Lq be its associated quasicen-tric linking system, and let � : Lq !F be the projection. Assume a compatible setof inclusions f�QPg has been chosen for Lq. Fix a �nite group � which is a p-group ora p0-group, and a surjective homomorphism

b� : �1(jLqj) �� :

Set � = b� Æj : S ��! �. Fix H � �, and set SH = ��1(H). Then (SH ;FH ;LH) is also ap-local �nite group, and (via the inclusion of LH into Lq) jLH j is homotopy equivalent

to the covering space of jLqj ' jLj with fundamental group b��1(H).

Proof. De�ne �: Mor(F q) !Sub(�) by setting �(�) = b�(��1(�)), where b� =

B(b�) ÆJ as above. Then � and � satisfy hypotheses (i){(iv) of De�nition 3.6: points (i)and (ii) follow from (A)q, while (iii) follows from Proposition 1.13 and (iv) from (C)q.Thus (�; �;�) is a fusion mapping triple on F q, and FH is a saturated fusion systemover SH by Proposition 3.8.

Let Op�(F) and O

p0

� (F) be the categories of De�nition 3.3, and let Op0

� (F)q � Op0

� (F)and Op

�(F)q � Op

�(F) be the full subcategories whose objects are the F -quasicentricsubgroups of S. By Proposition 3.8(b,c), �(L�1) contains O

p0

� (F)q (if � is a p0-group)

or Op�(F)

q (if � is a p-group). Hence in either case, by Lemma 3.4(b), all morphismsin Lq are composites of morphisms in L�1 and restrictions of morphisms in AutL(S).

Since L�1 = b��1(1), by de�nition, and since b�(�) = b�(�) whenever � is a restriction

of �, this shows that b� restricts to a surjection of AutL(S) onto �. In particular, thisimplies that

8 P 2 Ob(Lq) and 8 g 2 �, 9 P 0 � S and � 2 IsoLq(P; P0) such that b�(�) = g; (1)

where in fact, � can always be chosen to be the restriction of an automorphism of S.

We start by proving that LqH is a linking system associated to FH (for its set ofobjects), and hence that LH is a centric linking system. Let P � SH be a F -quasicentric

subgroup, and choose g 2 P �CS(P ). By construction, b�(ÆS(g)) = �(g), and b�(�P ) = 1.In particular, the inclusion morphisms are in LH . Also, �P Æ ÆP (g) = ÆS(g) Æ �P by

de�nition of an inclusion morphism (De�nition 1.11). Hence b�(ÆP (g)) = �(g) in thissituation; and in particular ÆP (g) 2 AutLq

H(P ) if and only if g 2 SH . Thus ÆP restricts

to a distinguished monomorphism P �CSH (P ) ��! AutLqH(P ) for LqH and axiom (D)q is

satis�ed. Moreover, if f; f 0 2 MorLqH(P;Q) are such that �(f) = �(f 0) in HomF(P;Q),and g 2 CS(P ) is the unique element such that f 0 = f Æ ÆP (g) (using axiom (A)q for

Lq), then b�(ÆP (g)) = 1, and hence g 2 SH . This shows that axiom (A)q for LqH holds.

Axioms (B)q and (C)q for LqH also follow immediately from the same properties for Lq.

We have now shown that (SH ;FH;LH) is a p-local �nite group. It remains to showthat jLH j is homotopy equivalent to a certain covering space of jLqj ' jLj. We showthis by �rst choosing certain full subcategories Lx � Lq and LxH � L

qH such that

jLxj ' jLqj ' jLj and jLxH j ' jLqH j ' jLH j, and then proving directly that jLxH j is a

covering space of jLxj.

If � is a p0-group, then for any P � SH = S, P is FH-centric if and only if it isF -centric (Proposition 3.8(c)). By the above remarks, LH is a centric linking systemassociated to FH . Set Lx = L and LxH = LH in this case.

If � is a p-group, then for any P � SH , P is FH-quasicentric if and only if it isF -quasicentric (Proposition 3.8(b)). So by what was just shown, LqH is a quasicentric


linking system associated to FH which extends LH . Let Lx � Lq be the full subcategorywhose objects are those subgroups P � S such that P \ SH is F -quasicentric. SetS1 = Ker(�), and let F1 � F be the saturated fusion system over S1 de�ned in

Proposition 3.8. The de�nition of � as a restriction of b� ensures that �(g) and �(g0)are �-conjugate whenever g; g0 are F -conjugate; in particular, no element of S1 is F -conjugate to any element of SrS1. Hence by Lemma 3.5, for each P � S which isF -centric and F -radical, P \ S1 is F1-centric, hence F -quasicentric, so P \ SH is F -quasicentric, and thus P 2 Ob(Lx). So the inclusion of jLxj in jLqj is a homotopyequivalence by Proposition 1.12.

Still assuming � is a p-group, let LxH � Lx be the subcategory with the same objects,

where Mor(LxH) =b��1(H). For each P 2 Ob(LxH) = Ob(Lx), P \SH is F -quasicentric

by assumption, hence FH-quasicentric; and by Proposition 1.13, each ' 2 MorLxH(P;Q)restricts to a unique morphism 'H 2 MorLxH(P \SH ; Q\SH). These restrictions de�nea deformation retraction from jLxHj to jL

qH j, and thus the inclusion of categories induces

a homotopy equivalence jLxHj ' jLqHj ' jLHj.

Thus in both cases, we have chosen categories LxH � Lx with the same objects,

where Lx is a full subcategory of Lq and Mor(LxH) = Mor(Lx) \ b��1(H), and wherejLxj ' jLj and jLxH j ' jLHj. Let E�(�=H) be the category with object set �=H,and with a morphism �g from aH to gaH for each g 2 � and aH 2 �=H. ThusAutE�(�=H)(1�H) �= H, and jE�(�=H)j = EG=H ' BH.

Let eL be the pullback category in the following square:

eL ! E�(�=H)

Lx#

! B(�) ,#

Thus Ob( eL) = Ob(Lx)� �=H, and Mor( eL) is the set of pairs of morphisms in Lx andE�(�=H) which get sent to the same morphism in B(�). Then LxH can be identi�ed

with the full subcategory of eL with objects the pairs (P; 1�H) for P 2 Ob(Lx). By (1),

each object in eL is isomorphic to an object in LxH , and so jLxHj ' j eLj. By construction,j eLj is the covering space over jLxj with fundamental group ��1(H). Since j eLj ' jLH jand jLxj ' jLj, this �nishes the proof of the last statement. �

The following is an immediate corollary to Theorem 3.9.

Corollary 3.10. For any p-local �nite group (S;F ;L), any �nite p-solvable group �,and any homomorphism b� : �1(jLj) ��! �;

there is a p-local �nite group (S0;F0;L0) such that jL0j is homotopy equivalent to the

covering space of jLj with fundamental group Ker(b�). Furthermore, this can be chosensuch that S0 is a subgroup of S and F0 is a subcategory of F .

For any p-local �nite group (S;F ;L), since F is a �nite category, Corollary 3.10implies that there is a unique maximal p-solvable quotient group of �1(jLj), which is�nite. In contrast, if we look at arbitrary �nite quotient groups of the fundamentalgroup, they can be arbitrarily large.

As one example, consider the case where p = 2, S 2 Syl2(A6) (so S �= D8), F =FS(A6), and L = LcS(A6). It is not hard to show directly, using Van Kampen's theorem,


that �1(jLj) �= �4�S�4: the amalgamated free product of two copies of �4 intersecting

in S, where each of the two subgroups C22 in D8 is normalized by one of the �4.

Thus �1(jLj) surjects onto any �nite group � which is generated by two copies of �4

intersecting in the same way. This is the case when � = A6, and also when � = PSL2(q)for any q � �9 (mod 16). However, the kernel of any such homomorphism de�ned on�1(jLj) is torsion free (and in�nite), and hence cannot be the fundamental group ofthe geometric realization of any centric linking system. In fact, in this case, there isno nontrivial homomorphism from �1(jLj) to a �nite 2-solvable group.

4. Fusion subsystems and extensions of p-power index

Recall that for any saturated fusion system F over a p-group S, we de�ned OpF(S) C

S to be the subgroup generated by all elements of the form g�1�(g), for g 2 P � S,and � 2 AutF(P ) of order prime to p. In this section, we classify all saturated fusionsubsystems of p-power index in a given saturated fusion system F over S, and showthat there is a one-to-one correspondence between such subsystems and the subgroupsof S which contain Op

F(S). In particular, there is a unique minimal subsystem Op(F)of this type, which is a fusion system over Op

F(S). We then look at extensions, anddescribe the procedure for �nding all larger saturated fusion systems of which F is afusion subsystem of p-power index.

4.1 Subsystems of p-power index In this subsection, we classify all saturatedfusion subsystems of p-power index in a given saturated fusion system F , and showthat there is a unique minimal subsystem Op(F) of this type. We also show that thereis a bijective correspondence between subgroups T of the �nite p-group

�p(F)def= S=Op

F(S);

and fusion subsystems FT of p-power index in F . We have already seen that �p(F) =S=Op

F(S) is isomorphic to �1(jLj^p ) for any centric linking system L associated to F ,

and our result also shows that all (connected) covering spaces of jLj^p are realized asclassifying spaces of subsystems of p-power index. The index of FT in F can thenbe de�ned to be the index of T in �p(F), or equivalently the covering degree of thecovering space.

The main step in doing this is to construct a fusion mapping triple (�p(F); �;�) forF q, where � is the canonical surjection of S onto �p(F). This construction parallelsvery closely the construction in Proposition 2.4 of a functor from Lq to B(�p(F)), whenL is a linking system associated to F . In fact, we could in principal state and provethe two results simultaneously, but the extra terminology which that would requireseemed to add more complications than would be saved by combining the two.

The following lemma provides a very general, inductive tool for constructing explicitfusion mapping triples.

Lemma 4.1. Fix a saturated fusion system F over a p-group S. Let H0 be a set ofF-quasicentric subgroups of S which is closed under F-conjugacy and overgroups. LetP be an F-conjugacy class of F-quasicentric subgroups maximal among those not inH0, set H = H0 [ P, and let FH0 � FH � F q be the full subcategories with these


objects. Fix a group � and a homomorphism � : S ! �, and let

�0 : Mor(FH0) ��! Sub(�)

be such that (�; �;�0) is a fusion mapping triple for FH0. Let P 2 P be fully normalizedin F , and �x a homomorphism

�P : AutF(P ) ��! N�(�(CS(P )))=�(CS(P ))

such that the following two conditions hold:

(+) x�(g)x�1 = �(�(g)) for all g 2 P , � 2 AutF (P ), and x 2 �P (�).

(�) For all P � Q � S such that P C Q and Q is fully normalized in NF(P ), and forall � 2 AutF(P ) and � 2 AutF (Q) such that � = �jP , �P (�) � �0(�).

Then there is a unique extension of �0 to a fusion mapping triple (�; �;�) for FH suchthat �(�) = �P (�) for all � 2 AutF(P ).

Proof. Note that (+) is just point (v) of Lemma 3.7 applied to the subgroup P , while(�) is just point (vi) applied to restrictions to P . So both of these conditions arenecessary if we want to be able to extend �0 and �P to a fusion mapping triple forFH.

The uniqueness of the extension is an immediate consequence of Alperin's fusiontheorem, in the form of Theorem 1.5(a). The proof of existence is almost identical tothe proof of Lemma 2.3, so we just sketch it here brie y.

We �rst show that we can replace (�) by the following (a priori stronger) statement:

(��) for all Q;Q0 � S which strictly contain P , and for all � 2 HomF(Q;Q0) and

� 2 AutF(P ) such that � = �jP , �P (�) � �0(�).

It suÆces to show this when P is normal in Q and Q0, since otherwise we can replaceQ and Q0 by NQ(P ) and NQ0(P ). In this case, � 2 HomNF (P )(Q;Q

0), and by Theorem1.5(a) (Alperin's fusion theorem), it is a composite of restrictions of automorphisms ofsubgroups fully normalized in NF(P ). So it suÆces to prove (��) when � is such anautomorphism, and this is what is assumed in (�).

Now �x any morphism ' 2 HomF (P1; Q) which lies in FH but not in FH0; thus P1 2

P. Set P2 = '(P1) � Q, and let '0 2 IsoF(P1; P2) be the \restriction" of '. By Lemma1.3, there are isomorphisms 'i 2 IsoFq(NS(Pi); Ni), for some Ni � NS(P ) containingP , which restrict to isomorphisms 'i 2 IsoFq(Pi; P ). Fix elements xi 2 �0('i). Set = '2 Æ '

0Æ '�11 2 AutFq(P ). Thus '0 = '�12 Æ Æ '1, and we de�ne

�(') = �('0) = x�12 ��P ( )�x1:

This is independent of the choice of xi, since �0('i) � �(CS(P ))�xi by axioms (i) and(ii) in the de�nition of a fusion mapping triple. It is independent of the choice of '1and '2 by the same argument as was used in the proof of Lemma 2.3 (and this is wherewe need point (��)). Conditions (i){(iv) are easily checked. For example, (iv) | thecondition that x�(g)x�1 = �(�(g)) whenever g 2 P1, ' 2 HomF(P1; P2), and x 2 �(')| holds when ' can be extended to a larger subgroup since (�; �;�0) is already afusion mapping triple, holds for ' 2 AutF(P ) by (+), and thus holds in the generalcase since �(') was de�ned via a composition of such morphisms. Thus (�; �;�) is afusion mapping triple on FH. �


The construction of a fusion mapping triple to �p(F) in the following lemma is a�rst application of Lemma 4.1. Another application will be given in the next section.

Lemma 4.2. Let F be a saturated fusion system over a p-group S, and let

� : S ��! �p(F) = S=OpF(S)

be the projection. Then there is a fusion mapping triple (�p(F); �;�) on F q.

Proof. The function � will be constructed inductively, using Lemma 4.1. Let H0 �Ob(F q) be a subset (possibly empty) which is closed under F -conjugacy and over-groups. Let P be an F -conjugacy class of F -quasicentric subgroups maximal amongthose not in H0, set H = H0 [ P, and let FH0 � FH � F q be the full subcate-gories with these objects. Assume we have already constructed a fusion mapping triple(�p(F); �;�0) for FH0 .

We recall the notation of Lemma 2.2. If G is any �nite group, and S 2 Sylp(G), then

OpG(S)

def=[g; x]

�� g 2 P � S; x 2 NG(P ) of order prime to p�:

By Lemma 2.2, OpG(S) = S \ Op(G), and hence G=Op(G) �= S=Op

G(S).

Fix P 2 P which is fully normalized in F . Let N0 be the subgroup generated bycommutators [g; x] for g 2 NS(P ) and x 2 NAutF (P )(NS(P )) of order prime to p. ThenAutS(P ) 2 Sylp(AutF(P )) and AutN0

(P ) = OpAutF (P )

(AutS(P )), and by Lemma 2.2,

AutF(P )ÆOp(AutF(P )) �= AutS(P )=AutN0

(P ) �= NS(P )ÆhN0; CS(P )i:

Also, N0 � OpF(S), and so the inclusion of NS(P ) into S induces a homomorphism

�P : AutF(P ) �� AutF(P )=Op(AutF (P ))

�=NS(P )=hN0;CS(P )i

��! N�p(F)(�(CS(P )))=�(CS(P ))�=NS(CS(P )�S0)=(CS(P )�S0)

:

Here, we write S0 = OpF(S) for short; thus �p(F) = S=S0 and �(CS(P )) = CS(P )�S0=S0.

Point (+) in Lemma 4.1 holds by the construction of �P . So it remains only to provethat condition (�) in Lemma 4.1 holds.

To see this, �x P � Q � S such that P C Q and Q is fully normalized in NF(P ),and �x � 2 AutFq(P ) and � 2 AutFq(Q) such that � = �jP . We must show that�P (�) � �0(�). Upon replacing � by �k and � by �k for some appropriate k � 1(mod p), we can assume that both automorphisms have order a power of p. Since Q isfully normalized, AutNS(P )(Q) is a Sylow subgroup of AutNF (P )(Q); and hence there areautomorphisms 2 AutF(Q) and 2 AutF(P ) of order prime to p such that = jPand � �1 = cgjQ for some g 2 NS(Q)\NS(P ). Then �

�1 = cgjP . Also, 1 2 �P ( )and 1 2 �0( ), since both automorphisms have order prime to p. So

�0(�) = �(cgjQ) = g��(CS(Q)) � g��(CS(P )) = �P (cgjP ) = �P (�):

Thus, by Lemma 2.3, we can extend �0 to a fusion mapping triple on FH. Uponcontinuing this procedure, we obtain a fusion mapping triple de�ned on all of F q. �

We now apply Lemma 4.2 to classify fusion subsystems of p-power index. Recallthat by de�nition, if F is a saturated fusion system over S and F0 � F is a fusionsubsystem over S0 � S, then F has p-power index if and only if S0 � Op

F(S), andAutF0(P ) � Op(AutF(P )) for all P � S0.

Theorem 4.3. Fix a saturated fusion system F over a p-group S. Then for eachsubgroup T � S containing Op

F(S), there is a unique saturated fusion system FT � Fover T with p-power index. For each such T , FT has the properties:


(a) a subgroup P � T is FT -quasicentric if and only if it is F-quasicentric; and

(b) for each pair P;Q � T of F-quasicentric subgroups,

HomFT (P;Q) =�' 2 HomF(P;Q)

��(') \ (T=OpF(S)) 6= ;

:

Here,

�: Mor(F q) ��! Sub(�p(F)) = Sub(S=OpF(S))

is the map of Lemma 4.2.

Proof. Let FT � F be the fusion system over T de�ned on F -quasicentric subgroupsby the formula in (b), and then extended to arbitrary subgroups by taking restrictionsand composites. (This is the fusion system denoted FT=Op

F(S) in Proposition 3.8, but

we simplify the notation here.) By Proposition 3.8(a,b) (applied with � = �p(F) andH = T=Op

F(S)), FT is saturated, a subgroup P � T is FT -quasicentric if and only if itis F -quasicentric, and AutFT (P ) � Op(AutF(P )) for all P � T .

Now let F 0T � F be another saturated subsystem over the same subgroup T which

also has p-power index. We claim that F 0T = FT , and thus that FT is the uniquesubsystem with these properties. By assumption, for each P � T , AutFT (P ) andAutF 0T (P ) both contain Op(AutF(P )), and hence each is generated by Op(AutF(P ))and any one of its Sylow p-subgroups. So if P is fully normalized in FT and F 0

T both,then

AutFT (P ) = hOp(AutF(P ));AutT (P )i = AutF 0T (P ): (1)

In particular, AutFT (T ) = AutF 0T (T ). Set pk = jT j, �x 0 � m < k, and assumeinductively that HomFT (P;Q) = HomF 0T

(P;Q) for all P;Q � T of order > pm. ByAlperin's fusion theorem for saturated fusion systems (Theorem 1.5(a)), if jP j = pm,jQj � pm, and P 6= Q, then all morphisms in HomFT (P;Q) and HomF 0T

(P;Q) arecomposites of restrictions of morphisms between subgroups of order > pm, and henceHomFT (P;Q) = HomF 0T

(P;Q) by the induction hypothesis. In particular, two sub-groups of order pm are FT -conjugate if and only if they are F 0T -conjugate. So for anyP � T of order pm, P is fully normalized in FT if and only if it is fully normalizedin F 0T . In either case, AutFT (P ) = AutF 0

T(P ): by (1) if P is fully normalized, and by

Alperin's fusion theorem again (and the induction hypothesis) if it is not. �

We can now de�ne Op(F) as the minimal fusion subsystem of F of p-power index:the unique fusion subsystem over Op

F(S) of p-power index. The next theorem will showthat when F has an associated linking system L, then Op(F) has an associated linkingsystem Op(L), and that jOp(L)j^p is the universal cover of jLj^p .

Theorem 4.4. Fix a p-local �nite group (S;F ;L). Then for each subgroup T � Scontaining Op

F(S), there is a unique p-local �nite subgroup (T;FT ;LT ) such that FT hasp-power index in F , and such that LqT = ��1(F qT ) where � is the usual projection of Lq

onto F q. Furthermore, jLT j is homotopy equivalent, via the inclusion of jLqT j ' jLT jinto jLqj ' jLj, to a covering space of jLj of degree [S : T ]. Hence the classifyingspace jLT j

^p of (T;FT ;LT ) is homotopy equivalent to the covering space of jLj^p with

fundamental group T=OpF(S).

Proof. Assume a compatible set of inclusions f�QPg has been chosen for Lq. By Proposi-tion 2.4, there is a functor � : Lq !B(�p(F)) which sends inclusions to the identity,and such that �(ÆS(g)) = g for all g 2 S. Hence by Theorem 3.9, (T;FT ;LT ) is a


p-local �nite group, and jLT j is a covering space of jLj. Also, if we write L1 for thelinking system over Op

F(S), then the �bration sequences

jL1j ! jLj ! B�p(F) and jL1j ! jLT j ! B(T=OpF(S))

are still �bration sequences after p-completion [BK, II.5.2(iv)], and hence jLT j^p is the

covering space of jLj^p with fundamental group T=OpF(S). The uniqueness follows from

Theorem 4.3. �

Thus there is a bijective correspondence between fusion subsystems of (S;F), orp-local �nite subgroups of (S;F ;L), of p-power index, and subgroups of S=Op

F(S)�=

�1(jLj^p ). The classifying spaces of the p-local �nite subgroups of (S;F ;L) of p-powerindex are (up to homotopy) just the covering spaces of the classifying space of (S;F ;L).

4.2 Extensions of p-power index We next consider the opposite problem: howto construct extensions of p-power index of a given p-local �nite group. In the courseof this construction, we will see that the linking system really is needed to constructan extension of the fusion system. The following de�nition will be useful.

De�nition 4.5. Fix a saturated fusion system F over a p-group S. An automorphism� 2 Aut(S) is fusion preserving if it normalizes F ; i.e., if it induces an automorphismof the category F by sending P to �(P ) and ' 2 Mor(F) to �'��1 2 Mor(F). LetAutfus(S;F) � Aut(S) denote the group of all fusion preserving automorphisms, andset

Outfus(S;F) = Autfus(S;F)=AutF(S):

We �rst describe the algebraic data needed to determine extensions of p-power index.Fix a p-local �nite group (S;F ;L), let Lq be the associated quasicentric linking system,and let f�QPg be a compatible set of inclusions. Then for any g 2 S, g acts on the setMor(Lq) by composing on the left or right with ÆS(g) and its restrictions. Thus for any' 2 MorLq(P;Q), we set

g' = ÆQ;gQg�1(g) Æ ' 2 MorLq(P; gQg�1)

and'g = ' Æ Æg�1Pg;P (g) 2 MorLq(g

�1Pg;Q):

This de�nes natural left and right actions of S on the set Mor(Lq). The resultingconjugation action ' 7! g'g�1 extends to an action on the category, where g sendsan object P to gPg�1. The functor � : Lq !F q is equivariant with respect to theconjugation action of S on Lq and the action of Inn(S) � Autfus(S;F) on F .

If (S0;F0;L0) is contained in (S;F ;L) with p-power index, and S0 C S, then the Saction on L clearly restricts to an S-action on L0. The following theorem provides aconverse to this. Given a p-local �nite group (S0;F0;L0), an extension S of S0, and anS-action on L which satis�es certain obvious compatibility conditions, this data alwaysdetermines a p-local �nite group which contains (S0;F0;L0) with p-power index.

Theorem 4.6. Fix a p-local �nite group (S0;F0;L0), and assume that a compatible

set of inclusions f�QPg has been chosen for L0. Fix a p-group S such that S0 C S andAutS(S0) � Autfus(S0;F0), and an action of S on L0 which:

(a) extends the conjugation action of S0 on L0;

(b) makes the canonical monomorphism ÆS0 : S0 ��! AutL0(S0) S-equivariant;


(c) makes the projection � : L0 ��! F0 S-equivariant with respect to the AutS(S0)-action on F0; and

(d) sends inclusion morphisms in L0 to inclusion morphisms.

Then there is a p-local �nite group (S;F ;L) such that F � F0, Lq � L0, the conju-gation action of S on Lq restricts to the given S-action on L0, and (S0;F0;L0) is asubgroup of p-power index in (S;F ;L).

Proof. Set

H0 = Ob(L0) = fP � S0 jP is F0-centricg and H = fP � S jP \ S0 2 H0g:

To simplify notation, for any P � S, we write P0 = P \ S0. For g 2 S and ' 2MorL0(P;Q), we write g'g

�1 2 MorL0(gPg�1; gQg�1) for the given action of g on '.

By (a), when g 2 S0, this agrees with the morphism g'g�1 already de�ned.

Step 1: We �rst de�ne categories L1 � L0 and F1 � F0, where Ob(F1) = Ob(F0)and Ob(L1) = H0. Set

Mor(L1) = S �S0 Mor(L0) =�S �Mor(L0)

�Æ�;

where (gg0; ') � (g; g0') for g 2 S, g0 2 S0, and ' 2 Mor(L0). If ' 2 MorL0(P;Q),then [[g; ']] 2 MorL1(P; gQg

�1) denotes the equivalence class of the pair (g; '). Com-position is de�ned by

[[g; ']] Æ [[h; ]] = [[gh; h�1'h Æ ]]:

Note that if ' 2 MorL0(P;Q), then h�1'h 2 MorL0(h

�1Ph; h�1Qh). To show that thisis well de�ned, we note that for all g; h 2 S, g0; h0 2 S0, and '; 2 Mor(L0) withappropriate domain and range,

[[gg0; ']] Æ [[hh0; ]] = [[gg0hh0; h�10 (h�1'h)h0 Æ ]] = [[gh�(h�1g0h); (h

�1'h) Æ h0 ]]

= [[gh; (h�1g0h)�(h�1'h) Æ h0 ]] = [[gh; h�1(g0')h Æ h0 ]] = [[g; g0']] Æ [[h; h0 ]] :

Here, the second to last equality follows from assumptions (b) and (d).

Let F1 be the smallest fusion system over S0 which contains F0 and AutS(S0). Byassumption, AutS(S0) � Autfus(S0;F0). Thus for each g 2 S, cg normalizes the fusionsystem F0: for each ' 2 Mor(F0) there is '0 2 Mor(F0) such that 'Æcg = cg Æ'

0. Henceeach morphism in F1 has the form cg Æ ' for some g 2 S and ' 2 Aut(F0). De�ne

�L1 : L1 ��! F1

by sending �L1([[g; ']]) = cg Æ �0('), where �0 denotes the natural projection from L0to F0. This is a functor by (c).

For all P;Q 2 H0, de�ne

bÆP;Q : NS(P;Q) ��! MorL1(P;Q)

by setting bÆP;Q(g) = [[g; �g�1QgP ]]. This extends the canonical monomorphism ÆP;Q de�ned

from NS0(P;Q) to MorL0(P;Q). To simplify the notation below, we sometimes write

bx = bÆP;Q(x) for x 2 NS(P;Q).

Step 2: We next construct categories L2 and F2, both of which have object sets H,and which contain L1 and the restriction of F1 to H0, respectively. Afterwards, we letF be the fusion system over S generated by F2 and restrictions of morphisms.


Before doing this, we need to know that the following holds for each P;Q 2 H0 andeach 2 MorL1(P;Q):

8 x 2 NS(P ) there is at most one y 2 NS(Q) such that by Æ = Æ bx: (1)

Since is the composite of an isomorphism and an inclusion (and the claim clearlyholds if is an isomorphism), it suÆces to prove this when P � Q and is theinclusion. We will show that we must have y = x in that case. By de�nition,

�QP Æ bx = [[1; �QP ]] Æ [[x; IdP ]] = [[x; �x�1QxP ]] and by Æ �QP = [[y; �QP ]]

(since conjugation sends inclusions to inclusions). So if by Æ �QP = �QP Æ bx, then there exists

g0 2 S0 such that y = xg0 and �QP = g�10 �x

�1QxP , and thus

ÆP;Q(1) = �QP = g�10 �x�1QxP = Æx�1Qx;Q(g

�10 ) Æ ÆP;x�1Qx(1) = ÆP;Q(g

�10 ):

Hence g0 = 1 by the injectivity of ÆP;Q in Proposition 1.13; and so by Æ �QP = �QP Æ bx onlyif x = y 2 NS(Q).

Now let L2 be the category with Ob(L2) = H, and where for all P;Q 2 H,

MorL2(P;Q) =� 2 MorL1(P0; Q0)

��8 x 2 P 9 y 2 Q such that Æ bx = by Æ :

Let bÆP;Q : NS(P;Q)�NS(P0;Q0)

��! MorL2(P;Q)�MorL1(P0;Q0)

be the restriction of bÆP0;Q0. Let F2 be the category with Ob(F2) = H, and where

MorF2(P;Q) =�' 2 Hom(P;Q)

�� 9 2 MorL2(P;Q) such that Æ bx =['(x) Æ 8x 2 P:

Let � : L2 !F2 be the functor which sends 2 MorL2(P;Q) to the homomorphism�( )(x) = y if Æ bx = by Æ (uniquely de�ned by (1)). Let F be the fusion system overS generated by F2 and restriction of homomorphisms.

Set � = S=S0. Let b� : L2 ��! B(�)be the functor de�ned by setting b�([[g; ']]) = gS0. In particular,

��1(1)

�jH0

= L0.

Step 3: We next show that each P 2 H is F -conjugate to a subgroup P 0 such thatP 00 is fully normalized in F0. Moreover, we show that P 0 can be chosen so that the

following holds:

8 g 2 S such that gP0g�1 is F0-conjugate to P0, g�S0 \NS(P

00) 6= ;. (2)

To see this, let Pfn be the set of all S0-conjugacy classes [P 00] of subgroups P

00 � S0

which are F0-conjugate to P0 and fully normalized in F0. (If P 00 is fully normalized in

F0, then so is every subgroup in [P 00].) Let S 0 � S be the subset of elements g 2 S

such that gP0g�1 is F0-conjugate to P0. In particular, S 0 � NS(P0) � P . Since each

g 2 S acts on F0 | two subgroups Q;Q0 � S0 are F0-conjugate if and only if gQg�1

and gQ0g�1 are F0-conjugate | S 0 is a subgroup of S.

For all g 2 S 0 and [P 00] 2 Pfn, gP

00g�1 is F0-conjugate to gP0g�1 and hence to P0,

and is fully normalized since g normalizes S0. Thus S0=S0 acts on Pfn, and this set has

order prime to p by Proposition 1.16. So we can thus choose a subgroup P 00 � S0 such

that [P 00] 2 Pfn and is �xed by S 0. In particular, P 0

0 is fully normalized in F0. Also, foreach g 2 S 0, some element of g�S0 normalizes P

00 (since [P

00] = [gP 0

0g�1]) | and this

proves (2).


Now consider the set RepF0(P0; S0) = HomF0(P0; S0)=Inn(S0). Since P � S 0, thegroup P=P0 acts on this set by conjugation (i.e., gP0 2 P=P0 acts on ['], for ' 2HomF0(P0; S0), by sending it to [cg'c

�1g ]). In particular, since the S0-conjugacy class

[P 00] is invariant under conjugation by P=P0, this group leaves invariant the subset X �

RepF0(P0; S0) of all conjugacy classes ['] of homomorphisms such that [Im(')] = [P 00].

Fix any ' 2 IsoF0(P0; P00) (recall that subgroups in Pfn are F0-conjugate to P0). Every

element of X has the form [�'] for some � 2 AutF0(P00), and [�'] = [�'] if and only if

��1 2 AutS0(P00). Thus jXj = jAutF0(P

00)jÆjAutS0(P

00)j, and is prime to p since P 0

0 isfully normalized in F0. We can thus choose '0 2 HomF0(P0; S0) such that '0(P0) = P 0

0

and ['0] is invariant under the P=P0-action.

Fix 2 IsoL0(P0; P00) such that �( ) = '0. Since ['0] is P=P0-invariant, for each

x 2 P , there is some y 2 x�S0 such that cy Æ '0 = '0 Æ cx. By (A)q (and since is an

isomorphism), there is y0 2 y�CS0(P00) such that by0 Æ = Æ bx in L1. This element y0 is

unique by (1); and upon setting '(x) = y0 we get a homomorphism ' 2 HomF (P; S)which extends '0. Set P

0 = '(P ); then P 0 is F -conjugate to P and P 00 = P 0 \ S0.

Step 4: In Step 5, we will prove that F is saturated, using [5A1, Theorem 2.2].Before that theorem can be applied, a certain technical condition must be checked.

Assume that P is F -centric, but not in H. By Step 3, P is F -conjugate to someP 0 such that P 0

0 is fully normalized in F0. Thus P00 is fully centralized in F0 and not

F0-centric, which implies that CS0(P00) � P 0

0. Then P0 acts on CS0(P

00)�P

00=P

00 with �xed

subgroup QP 00=P

00 6= 1 for some Q � CS0(P

00), and [Q;P 0] � P 0

0 since P0=P 0

0 centralizesQP 0

0=P00. Hence Q � P 0

0, and Q � NS(P0) since [Q;P 0] � P 0. For any x 2 QrP 0

0,[cx] 6= 1 2 Out(P 0) (CS(P

0) � P 0 since P is F -centric), but cx induces the identity onP 00 (since Q � CS0(P

00)) and on P 0=P 0

0 (since x 2 S0). Hence [cx] 2 Op(OutF (P0)) by

Lemma 1.15. This shows that

P F -centric, P =2 H =) 9P 0 F -conjugate to P , OutS(P0)\Op(OutF(P

0)) 6= 1: (3)

Step 5: We next show that F is saturated, and also (since it will be needed in theproof of (II)) that axiom (A)q holds for L2. By [5A1, Theorem 2.2] (the stronger formof Theorem 1.5(b)), it suÆces to prove that the subgroups in H satisfy the axioms forsaturation. Note in particular that condition (�) in [5A1, Theorem 2.2] is preciselywhat is shown in (3).

Proof of (I): Fix a subgroup P 2 H which is fully normalized in F . Let S 0=S0 �S=S0 be the stabilizer of the F0-conjugacy class of P0. By Step 3, P is F -conjugate toa subgroup P 0 such that P 0

0 is fully normalized in F0; and such that for each g 2 S 0,some element of g�S0 normalizes P 0

0 (see (2)). Hence there are short exact sequences

1 ��! AutL0(P00) ��! AutL1(P

00) ��! S 0=S0 ��! 1

1 ��! NS0(P00) ��! NS(P

00) ��! S 0=S0 ��! 1:

We consider P 0 � NS(P00) as subgroups of AutL1(P

00) via

bÆP 00;P 0

0. Then

[AutL1(P00) : NS(P

00)] = [AutL0(P

00) : NS0(P

00)]

is prime to p since P 00 is fully normalized in F0, and hence NS(P

00) 2 Sylp(AutL1(P

00)).

Fix 2 AutL1(P00) such that �1NS(P

00) contains a Sylow p-subgroup of the group

NAutL1(P00)(P

0) (in particular, P 0 � �1NS(P00) ), and set P 00 = P 0 �1 � NS(P

00).

Then 2 IsoL2(P0; P 00) by de�nition of L2. In particular, P 00 is F -conjugate to P , and


P 000 = P 0

0. Also, NS(P00) contains a Sylow p-subgroup of NAutL1(P

00)(P

00),

AutL2(P00) = NAutL1(P

00)(P

00) and NS(P00) = NNS(P

00)(P

00);

and it follows that NS(P00) 2 Sylp(AutL2(P

00)).

Now, AutL2(P )�= AutL2(P

00) since they are F -conjugate, and jNS(P )j � jNS(P00)j

since P is fully normalized. Thus NS(P ) 2 Sylp(AutL2(P )), and hence AutS(P ) 2Sylp(AutF(P )). Also, NS(P ) contains the kernel of the projection from AutL2(P ) toAutF (P ); i.e., CS(P ) is isomorphic to this kernel. For all Q which is F -conjugate toP , CS(Q) is isomorphic to a subgroup of the same kernel, so jCS(Q)j � jCS(P )j, andthus P is fully centralized in F .

Proof of (A)q for L2: It is clear from the construction that for any P;Q 2 H,

CS(P ) acts freely on MorL2(P;Q) via bÆP;P . So it remains to show that when P is fullycentralized in F , then for all ; 0 2 MorL2(P;Q) such that �( ) = �( 0), there issome x 2 CS(P ) such that 0 = Æ bx. Since every morphism in L2 is the compositeof an isomorphism followed by an inclusion, it suÆces to show this when and 0

are isomorphisms. But in this case, �1 0 2 AutL2(P ) lies in the kernel of the mapto AutF(P ). We have just seen, in the proof of (I), that this implies there is somex 2 CS(P ) such that bx = �1 0, so 0 = Æ bx, and this is what we wanted to prove.

Proof of (II): Fix ' 2 HomF(P; S), where '(P ) is fully centralized in F . SetP 0 = '(P ). By de�nition of F2 � F and of L2, there is some 2 IsoL2(P; P

0) �

IsoL1(P0; P00) such that Æ bg = d'(g) Æ for all g 2 P . Upon replacing by bx Æ for

some appropriate x 2 S (and replacing ' by cx Æ ' and P 0 by xP 0x�1), we can assumethat 2 IsoL0(P0; P

00) and 'jP0 2 HomF0(P0; S0).

Consider the subgroups

N' = fx 2 NS(P ) j'cx'�1 2 AutS(P

0)g

N 0 = N'jP0 \N' =�x 2 N' \ S0

�� ('cx'�1)jP0 2 AutS0(P00):

We will see shortly that N 0 = N' \ S0. By (II) applied to the saturated fusion system

F0, there is '0 2 HomF0(N0; S) which extends 'jP0, and it lifts to 2 HomL0(N

0; S).

By (A) (applied to L0), there is z 2 Z(P0) such that = ( jP0) Æ bz. Upon replacing

by Æ bz (and '0 by '0 Æ cz), we can assume that = jP0.

For any x 2 N' \ S0, 'cx'�1 = cy 2 AutF(P0) for some y 2 NS(P

0). Hence by(A)q (for L2), bx �1 =cyz for some unique z 2 CS(P 0). Thus cyz 2 AutL0(P

00), and by

de�nition of the distinguished monomorphisms for L1, this is possible only if yz 2 S0.Thus 'cx'

�1 = cyz where yz 2 NS0(P0), and so x 2 N 0. This shows that N 0 = N'\S0.

De�ne ' 2 Hom(N'; S) by the relation '(x) = y if bx �1 = by. In particular, this

implies that[yx�1 = Æ(x x�1)�1 is a morphism in L0, and hence that yx�1 2 S0. Also,'jP = ' by the original assumption on . It remains to show that ' 2 HomF(N'; S).

To do this, it suÆces to show that Æ bx = by Æ in MorL2(N0; S) for all x, where

y = '(x). Equivalently, we must show that

=[yx�1 Æ (x x�1): (4)

Since yx�1 2 S0, both sides in (4) are in L0, and they are equal after restriction to P0.Hence they are equal as morphisms de�ned on N' \ S0 by [5A1, Lemma 3.9], and this�nishes the proof.


Step 6: We next check that F0 has p-power index in F . For any P � S and any� 2 AutF(P ) of order prime to P , � induces the identity on P=P0 by construction,and hence x�1�(x) 2 S0 for all x 2 P . This shows that S0 � Op

F(S) (see De�nition2.1). Also, by construction, for all P � S0, AutF0(P ) is normal of p-power index inAutF (P ), and thus contains Op(AutF(P )). This proves that the fusion subsystem F0has p-power index in F in the sense of De�nition 3.1.

Step 7: It remains to construct a quasicentric linking system Lq which containsL2 as a full subcategory, and which is associated to F . Note �rst that the axioms ofDe�nition 1.9 are all satis�ed by L2: axiom (A)q holds by Step 5, while axioms (B)q,(C)q, and (D)q follow directly from the construction in Steps 1 and 2.

Let Lc2 � L2 be the full subcategory whose objects are the set Hc � H of subgroups

in H which are F -centric. We �rst construct a centric linking system L � Lc2 associ-ated to F . For any set K of F -centric subgroups of S, let Oc(F) be the orbit categoryof F , let OK(F) � Oc(F) be the full subcategory with object set K, and let ZKF bethe functor P 7! Z(P ) on OK(F) which sends (see De�nition 1.7 for more detail).By [BLO2, Proposition 3.1], when K is closed under F -conjugacy and overgroups, theobstruction to the existence of a linking system with object set K lies in lim �

3(ZKF ), and

the obstruction to its uniqueness lies in lim �2(ZKF ). Furthermore, by [BLO2, Proposi-

tion 3.2], if P is an F -conjugacy class of F -centric subgroups maximal among thosenot in K, and K0 = K [ P, then the inclusion of functors induces an isomorphismbetween the higher limits of ZKF and ZK

0

F if certain groups ��(OutF (P );Z(P )) vanishfor P 2 P. By (3), Op(OutF(P )) 6= 1 for any F0-centric subgroup P =2 Hc, and hence��(OutF(P );Z(P )) = 0 for such P by [JMO, Proposition 6.1(ii)]. So these obstructionsall vanish, and there is a centric linking system L � Lc2 associated to F .

Now let Lq be the quasicentric linking system associated to (S;F ;L). For eachP 2 H = Ob(L2), P \ S0 is F0-centric by de�nition, hence is F -quasicentric byTheorem 4.3(a), and thus P is also F -quasicentric. Also, H contains all F -centricF -radical subgroups by (3), and H is closed under F -conjugacy and overgroups (byde�nition of F and H). Hence by [5A1, Proposition 3.12], since Lc2 is a full subcategoryof L by construction, L2 is isomorphic to a full subcategory of Lq in a way whichpreserves the projection functors and distinguished monomorphisms. So L0 can alsobe identi�ed with a linking subsystem of Lq, and this �nishes the proof. �

We now prove a topological version of Theorem 4.6. By Theorem 4.4, if (S0;F0;L0) �(S;F ;L) is an inclusion of p-local �nite groups of p-power index, where S0 C S and� = S=S0, then there is a �bration sequence jL0j

^p ��! jLj

^p ��! B�. So it is natural

to ask whether the opposite is true: given a �bration sequence whose base is theclassifying space of a �nite p-group, and whose �ber is the classifying space of a p-local�nite group, is the total space also the classifying space of a p-local �nite group? Thenext proposition shows that this is, in fact, the case.

Before stating the proposition, we �rst de�ne some categories which will be needed inits proof. Fix a space Y , a p-group S, and a map f : BS ��! Y . For P � S, we regardBP as a subspace of BS; all of these subspaces contain the basepoint � 2 BS. Wede�ne three categories in this situation, FS;f(Y ), LS;f(Y ), andMS;f(Y ), all of whichhave as objects the subgroups of S. Of these, the �rst two are discrete categories, whileMS;f(Y ) has a topology on its morphism sets. Morphisms in FS;f(Y ) are de�ned bysetting

MorFS;f (Y )(P;Q) =�' 2 Hom(P;Q)

�� f jBP ' f jBQ Æ B';


we think of this as the fusion category of Y (with respect to S and f).

Next de�ne

MorMS;f (Y )(P;Q) =�(';H)

��' 2 Hom(P;Q), H : BP � [0; t] ! Y ,

t � 0, HjBP�0 = f jBP , HjBP�t = f jBQ Æ B':

Thus a morphism in MS;f(Y ) has the form (';H) where H is a Moore homotopy inY . Composition is de�ned by

( ;K) Æ (';H) = ( '; (K Æ (B'� Id))�H);

where if H and K are homotopies parameterized by [0; t] and [0; s], respectively, then(K Æ (B'� Id))�H is the composite homotopy parameterized by [0; t+ s].

Let P(Y ) be the category of Moore paths in Y , and let Res� :MS;f(Y ) !P(Y )be the functor which sends each object to f(�), and sends a morphism (';H) to thepath obtained by restricting H to the basepoint � 2 BS. De�ne a map ev from jP(Y )jto Y as follows. For any n-simplex �n in jP(Y )j, indexed by a composable sequenceof paths �1; : : : ; �n where �i is de�ned on the interval [0; ti], let evj�n be the composite

�n �(t1;:::;tn)��! [0; t1 + : : :+ tn]

�n��1��! Y;

where �(t0; : : : ; tn) is the aÆne map which sends the i-th vertex to t1 + : : : + ti. ThecategoryMS;f(Y ) thus comes equipped with an \evaluation function"

eval : jMS;f(Y )jjRes� j��! jP(Y )j

ev��! Y:

Now setMorLS;f (Y )(P;Q) = �0

�MorMS;f (Y )(P;Q)

�:

We think of LS;f(Y ) as the linking category of Y . Also, for any set H of subgroups ofS, we let LHS;f(Y ) andM

HS;f(Y ) denote the full subcategories of LS;f(Y ) andMS;f(Y )

with object set H. For more about the fusion and linking categories of a space, see[BLO2, x7].

Theorem 4.7. Fix a p-local �nite group (S0;F0;L0), a p-group �, and a �bration

Xv��! B� with �ber X0 ' jL0j

^p . Then there is a p-local �nite group (S;F ;L) such

that S0 C S, F0 � F is a fusion subsystem of p-power index, S=S0 �= �, and X ' jLj^p .

Proof. Let � denote the base point of B�, and assume X0 = v�1(�). Fix a homo-topy equivalence f : jL0j^p ��! X0, regard BS0 as a subspace of jL0j, and set f0 =f jBS0 : BS0 ��! X0, also regarded as a map to X. Let H0 be the set of F0-centricsubgroups of S0.

Step 1: By [BLO2, Proposition 7.3],

F0 �= FS0;f0(X0) and L0 �= LH0

S0;f0(X0):

We choose the inclusions �QP 2 MorL0(P;Q) (for P � Q in H0) to correspond to the

morphisms (inclQP ; [c]) in LS0;f0(X0), where c is the constant homotopy f0jBP . Set

F1 = FS0;f0(X) and L1 = LH0

S0;f0(X);

where f0 is now being regarded as a map BS0 ��! X. The inclusion X0 � X makesF0 into a subcategory of F1 and L0 into a subcategory of L1.

For all P � S0 which is fully centralized in F0, Map(BP;X0)f0jBP ' jCL0(P )j^p by

[BLO2, Theorem 6.3], where CL0(P ) is a linking system over the centralizer CS(P ).


Since P 2 H0 (i.e., P is F0-centric) if and only if it is fully centralized and CS0(P ) =Z(P ), this shows that

P 2 H0 () Map(BP;X0)f0jBP ' BZ(P ). (1)

If P and P 0 are F1-conjugate, and ' 2 IsoF1(P; P0), then the homotopy between f0jBP

and f0jBP 0 ÆB' as maps from BP to X induces, using the homotopy lifting property forthe �bration v, a homotopy between wÆf0jBP and f0jBP 0 ÆB' (as maps from BP toX0),

where w : X0'!X0 is the homotopy equivalence induced by lifting some loop in B�.

Since w is a homotopy equivalence and B' is a homeomorphism, this shows that themapping spaces Map(BP;X0)f0jBP and Map(BP 0; X0)f0jBP 0 are homotopy equivalent,and hence (by (1)) that P 0 2 H0 if P

0 2 H0. Thus for all P; P0 � S0,

P F1-conjugate to P0 and P 2 H0 =) P 0 2 H0. (2)

Fix S 2 Sylp(AutL1(S0)). We identify S0 as a subgroup of S via the distinguishedmonomorphism ÆS0 from S0 to AutL0(S0) � AutL1(S0).

Step 2: For all P � S, and for Y = X0 or X, we de�ne

Map(BP; Y )� =�f : BP ! Y

�� f ' f0 ÆB'; ' 2 Hom(P; S); '(P ) 2 H0

:

Using (2), we see that the �bration sequence X0 ��! X ��! B� induces a �brationsequence of mapping spaces

Map(BP;X0)� ��! Map(BP;X)� ��! Map(BP;B�)ct ' B�; (3)

where Map(BP;B�)ct is the space of null homotopic maps, and the last equivalenceis induced by evaluation at the basepoint. By (1), each connected component ofMap(BP;X0)� is homotopy equivalent to BZ(P ), and hence the connected compo-nents of Map(BP;X)� are also aspherical.

For any morphism ('; [H]) 2 MorL1(P;Q), where ' 2 HomF1(P;Q) and [H] is thehomotopy class of the path H in Map(BP;X)�, restricting v Æ H to the basepoint ofBP de�nes a loop in B�, and thus an element of �. This de�nes a map from Mor(L1)to � which sends composites to products, and thus a functor

b� : L1 ��! B(�):By construction (and the �bration sequence (3)), for any 2 Mor(L1), we have 2

Mor(L0) if and only if b�( ) = 1.

Using the homotopy lifting property in (3), we see that b� restricts to a surjectionof AutL1(S0) onto �, with kernel AutL0(S0). Since S0 2 Sylp(AutL0(S0)), this shows

that b� induces an isomorphism S=S0 �= �, where S is a Sylow p-subgroup of AutL1(S0)

which contains S0. Hence for any 2 Mor(L1), there is g 2 S such that b�( ) = b�(g);and for any such g there is a unique morphism 0 2 Mor(L0) such that = Æ(g) Æ 0where Æ(g) 2 Mor(L1) denotes the appropriate restriction of g 2 AutL1(S0). In otherwords,

Mor(L1) �= S �S0 Mor(L0): (4)

Step 3: The conjugation action of S on Mor(L0) � Mor(L1) de�nes an action of S onL0, which satis�es the hypotheses of Theorem 4.6. (Note in particular that this actionsends inclusions to inclusions, since they are assumed to be represented by constanthomotopies.) So we can now apply that theorem to construct a p-local �nite group(S;F ;L) which contains (S0;F0;L0) with p-power index. Let Lq be the quasicentriclinking system associated to (S;F ;L). By (4), the category L1 de�ned here is equal


to the category L1 de�ned in the proof of Theorem 4.6; i.e., the full subcategory of Lq

with object set H0.

Let H be the set of subgroups P � S such that P \S0 2 H0, and let L2 � Lq be thefull subcategory with object set H. By Step 4 in the proof of Theorem 4.6, all F -centricF -radical subgroups of S lie in H, and so jLj ' jLqj ' jL2j by Proposition 1.12(a).Also, jL1j is a deformation retract of jL2j, where the retraction is de�ned by sendingP 2 H to P \ S0 2 H0 (and morphisms are sent to their restrictions, uniquely de�nedby Proposition 1.12(b)). Thus jLj ' jL1j. So by Theorem 4.4, we have a homotopy�bration sequence jL0j

^p ! jL1j

^p ! B�.

Step 4: It remains to construct a homotopy equivalence jL1j^p ��! X, which extendsto a homotopy equivalence between the �bration sequences. This is where we need touse the topological linking categories de�ned above. Set

M0 =MH0

S0;f0(X0) and M1 =M

H0

S0;f0(X)

for short, and consider the following commutative diagram:

jL0j^p

proj0

'jM0j

^p

eval0!X0

jL1j^p

#

proj1

'jM1j

^p

#eval1!X .#

The vertical maps in the diagram are all inclusions. Also, X0 is p-complete by de�nitionand X by [BK, II.5.2(iv)], so the evaluation maps de�ned above extend to the p-completed nerves jMij

^p . The maps proj1 and eval1 both commute up to homotopy

with the projections to B�. The projection maps proj0 and proj1 are both homotopyequivalences: the connected components of the morphism spaces inMi are contractiblesince the connected components of the �ber and total space in (3) are aspherical (seeStep 2).

We claim that eval0 is homotopic to f Æproj0 as maps to X0. By naturality, it suÆcesto check this on the uncompleted nerve jM0j, and in the case where X0 = jLj

^p and

f = Id. But in this case, the only real di�erence between the maps is that eval0 sendsall vertices of jM0j to the base point of X0 = jLj

^p , while proj0 sends the vertex for a

subgroup P � S to the corresponding vertex in jLj^p . So the maps are homotopic via ahomotopy which sends vertices to the base point along the edges of jLj correspondingto the inclusion morphisms. In particular, this shows that eval0 is also a homotopyequivalence. We thus have an equivalence between the �bration sequences

jL0j^p ! jL1j

^p ! B�

jL0j^p

'#

!X#

! B� ,

wwwww

and this proves that the two sequences are equivalent. �

Recall (De�nition 4.5) that for any saturated fusion system F over a p-group S,Autfus(S;F) denotes the group of all fusion preserving automorphisms of S. The fol-lowing corollary to Proposition 4.6 describes how \exotic" fusion systems could poten-tially arise as extensions of p-power index; we still do not know whether the situationit describes can occur.


Corollary 4.8. Fix a �nite group G, with Sylow subgroup S 2 Sylp(G). Assumethere is an automorphism � 2 Autfus(S;FS(G)) of p-power order, which is not therestriction to S of an automorphism of G, and which moreover is not the restriction ofan automorphism of G0 for any �nite group G0 with S 2 Sylp(G

0) and FS(G0) = FS(G).

Then there is a saturated fusion system (bS; bF) � (S;FS(G)), such that FS(G) has p-

power index in bF, and such that bF is not the fusion system of any �nite group.

Proof. By [BLO1, Theorem E], together with [O1, Theorem A] and [O2, Theorem A],there is a short exact sequence

0 ��! lim �1

OcS(G)

(ZG) ��! Outtyp(LcS(G)) ��! Outfus(S;FS(G)) ��! 0

where Outfus(S;F) = Autfus(S;F)=AutF(S), where OcS(G) and ZG are the category

and functor of De�nition 1.7(b), and where Outtyp(LcS(G)) is the group of \isotypi-cal" automorphisms of LcS(G) modulo natural isomorphism (see the introduction of[BLO1], or [BLO1, De�nition 3.2], for the de�nition). Let [�] be the class of � inOutfus(S;FS(G)); then [�] lifts to an automorphism � of the linking system LcS(G).Upon replacing � by some appropriate power, we can assume that it still has p-powerorder. We can also assume, upon replacing � by another automorphism in the sameconjugacy class if necessary, that the �-action on AutL(S) leaves invariant the subgroupÆS(S); i.e., that the action of � on LcS(G) restricts to an action on S.

Set bS = Sohxi, where jxj = j�j and x acts on S via �. Then bS has an action onLcS(G) induced by the actions of S and of �, and this action satis�es conditions (a){(d)

in Theorem 4.6. Let bF � FS(G) be the saturated fusion system over bS constructed bythe theorem.

We claim that bF is not the fusion system of any �nite group. Assume otherwise:assume bF is the fusion system of a group bG. Since F has p-power index in bF , S �Op

bF(bS), and so S � bS \ Op( bG) by the hyperfocal subgroup theorem (Lemma 2.2). Set

G0 = S�Op( bG). Then G0 C bG since S C bS; bG=G0 �= bS=S 0, and thus G0 C bG has p-powerindex and S 2 Sylp(G

0). By Theorem 4.3, there is a unique saturated fusion subsystem

over S of p-power index in bF , and thus F = FS(G0). Also, x 2 bS � bG acts on G0 via anautomorphism whose restriction to S is �, and this contradicts the original assumptionabout �. �

5. Fusion subsystems and extensions of index prime to p.

In this section, we classify all saturated fusion subsystems of index prime to p in agiven saturated fusion system F , and show that there is a unique minimal subsystemOp0(F) of this type. More precisely, we show that there is a certain �nite group of orderprime to p associated to F , denoted below �p0(F), and a one-to-one correspondencebetween subgroups T � �p0(F) and fusion subsystems FT of index prime to p in F .The index of FT in F can then be de�ned to be the index of T in �p0(F).

Conversely, we also describe extensions of saturated fusion systems of index primeto p. Once more the terminology requires motivation. Roughly speaking, an extensionof index prime to p of a given saturated fusion system F is a saturated fusion systemF 0 over the same p-group S, but where the morphism set has been \extended" by an


action of a group of automorphisms whose order is prime to p. Here again, a one-to-onecorrespondence statement is obtained, thus providing a full classi�cation.

5.1 Subsystems of index prime to p We �rst classify all saturated fusion sub-systems of index prime to p in a given saturated fusion system F . The fusion sub-systems and associated linking systems will be constructed using Proposition 3.8 andTheorem 3.9, respectively. More precisely, Theorem 3.9 has already told us that forany p-local �nite group (S;F ;L), any surjection � of �1(jLj) onto a �nite p0-group �,and any subgroup H � �, there is a p-local �nite subgroup (S 0;F 0;L0) such that jL0jis homotopy equivalent to the covering space of jLj with fundamental group ��1(H).What is new in this section is �rst, that we describe the \universal" p0-group quotientof �1(jLj) as a certain quotient group of OutF(S); and second, we show that all fusionsubsystems of index prime to p in F (in the sense of De�nition 3.1) are obtained inthis way.

When applying Proposition 3.8 to this situation, we need to consider fusion mappingtriples (�; �;�) on F q, where � is �nite of order prime to p. Since � 2 Hom(S;�), itmust then be the trivial homomorphism. In this case, conditions (i){(iii) in De�nition

3.6 are equivalent to requiring that there is some functor b�: F q !B(�) such that

�(') = fb�(')g for each ' 2 Mor(F q) (and condition (iv) is redundant). So insteadof explicitly constructing fusion mapping triples, we instead construct functors of thisform.

We start with some de�nitions. For a �nite group G, one de�nes Op0(G) to bethe smallest normal subgroup of G of index prime to p, or equivalently the subgroupgenerated by elements of p-power order in G. These two de�nitions are not, in general,equivalent in the case of an in�nite group, (the case G = Z being an obvious example).We do, however, need to deal with such subgroups here. The following de�nition ismost suitable for our purposes, and it is a generalization of the �nite case.

De�nition 5.1. For any group G (possibly in�nite), let Op0(G) be the intersection ofall normal subgroups in G of �nite index prime to p.

In particular, under this de�nition, an epimorphism � : G �� H with Ker(�) �Op0(G) induces an isomorphism G=Op0(G) �= H=Op0(H). Thus by Proposition 2.6, forany p-local �nite group (S;F ;L), the projections of jLj ' jLqj onto jF cj and jF qjinduce isomorphisms

�1(jLj)ÆOp0(�1(jLj)) �= �1(jF

cj)ÆOp0(�1(jF

cj)) �= �1(jFqj)ÆOp0(�1(jF

qj)):

Fix a saturated fusion system F over a p-group S, and de�ne

�p0(F) = �1(jFcj)=Op0(�1(jF

cj)):

We will show that the natural functor

"Fc : F c ��! B(�p0(F))

induces a bijective correspondence between subgroups of �p0(F) and fusion subsystemsof F of index prime to p.

Recall (De�nition 3.3) that for any saturated fusion system F over a p-group S,Op0

� (F) � F is the smallest fusion subsystem which contains the groups Op0(AutF(P ))


for all P � S; i.e., the smallest fusion subsystem which contains all automorphisms inF of p-power order. De�ne

Out0F(S) =� 2 OutF(S)

��jP 2 MorOp

0

� (F)(P; S); some F -centric P � S

�:

Then Out0F(S) C OutF(S), since Op0

� (F) is normalized by AutF(S) (Lemma 3.4(a)).

Proposition 5.2. There is a unique functor

b� : F c ��! B(OutF(S)=Out0F(S))with the following properties:

(a) b�(�) = � (modulo Out0F(S)) for all � 2 AutF(S).

(b) b�(') = 1 if ' 2 Mor(Op0

� (F)c). In particular, b� sends inclusion morphisms to the

identity.

Furthermore, there is an isomorphism

� : �p0(F) = �1(jFcj)=Op0(�1(jF

cj))�=

��! OutF(S)=Out0F(S)

such that b� = B� Æ "cF .Proof. By Lemma 3.4(c), each morphism in F c factors as the composite of the restric-tion of a morphism in AutF(S) followed by a morphism in Op0

� (F)c. If

' = '1 Æ �1jP = '2 Æ �2jP ;

where ' 2 HomF (P;Q), �i 2 AutF(S), and 'i 2 HomOp

0

� (F)c(�i(P ); Q), then we can

assume (after factoring out inclusions) that all of these are isomorphisms, and hence

(�2 Æ ��11 )jP = '�12 Æ '1 2 Iso

Op0

� (F)c(�1(P ); �2(P )):

Thus �2 Æ ��11 2 Out0F(S); and so we can de�ne

b�(') = [�1] = [�2] 2 OutF(S)=Out0F(S):

This shows that b� is well de�ned on morphisms (and sends each object in F c to the

unique object of B(OutF(S)=Out0F(S))). By Lemma 3.4(c) again, b� preserves compo-

sitions, and hence is a well de�ned functor. It satis�es conditions (a) and (b) above by

construction. The uniqueness of b� is clear.It remains to prove the last statement. Since OutF(S)=Out

0F (S) is �nite of order

prime to p, �1(jb�j) factors through a homomorphism

�� : �1(jFcj)=Op0(�1(jF

cj)) ��! OutF (S)=Out0F(S):

The inclusion of B AutF(S) into jF cj (as the subcomplex with one vertex S) inducesa homomorphism

� : OutF(S) ��! �1(jFcj)=Op0(�1(jF

cj)):

Furthermore, � is surjective since F = hOp0

� (F);AutF(S)i (Lemma 3.4(b)), and sinceany automorphism in Op0

� (F) is a composite of restrictions of automorphisms of p-powerorder. By (a), and since � restricted to AutF(S) is the projection onto OutF(S), the

composite �Æ� is the projection of OutF(S) onto the quotient group OutF (S)=Out0F(S).

Finally, Out0F(S) � Ker(�) by de�nition of Out0F(S), and this shows that � is anisomorphism. �


The following lemma shows that any fusion mapping triple on F c can be extendeduniquely to F q. This will allow us later to apply Proposition 3.8 in order to producesaturated fusion subsystems of index prime to p.

Lemma 5.3. Let F be a saturated fusion system over a p-group S, and let (�; �;�0)be a fusion mapping triple on F c. Then there is a unique extension

�: Mor(F q) !Sub(�)

of �0, such that (�; �;�) is a fusion mapping triple on F q.

Proof. We construct the extension � one F -conjugacy class at a time. Thus, assume� has been de�ned on a set H0 of F -quasicentric subgroups of S which is a union ofF -conjugacy classes, contains all F -centric subgroups, and is closed under overgroups.Let P be maximal among F -conjugacy classes of F -quasicentric subgroups not in H0;we show that � can be extended to H = H0 [ P.

Fix P 2 P which is fully normalized in F . For each � 2 AutF(P ), there is anextension � 2 AutF(P �CS(P )) (axiom (II)), and we de�ne a map

�P : AutF(P ) !Sub(N�(�(CS(P ))))

by �P (�) = �(�)��(CS(P )). By De�nition 3.6 ((i) and (ii)) �(�) is a left coset of�(CS(P �CS(P ))), and by (iv), it is also a right coset (where the right coset representa-tive can be taken to be the same as the one representing the left coset); hence �P (�)is a left and a right coset of �(CS(P )) (again with the same coset representative onboth sides). If �0 2 AutF (P �CS(P )) is any other extension of �, then by [5A1, Lemma3.8], there is some g 2 CS(P ) such that �0 = cg Æ �. Thus, by De�nition 3.6 again,�(�0) = �(cg Æ �) = �(g)�(�), and

�(�0) � �(CS(P )) = �(g)�(�) � �(CS(P )) = �(�)�(�(g)) � �(CS(P )) = �(�) � �(CS(P ));

and so the de�nition of �P (�) is independent of the choice of the extension �. Thisshows that �P is well de�ned.

Notice also that �P clearly respects compositions, and since �P (�) = x ��(CS(P )) =�(CS(P )) � x, for some x 2 �, we conclude that x 2 N�(�(CS(P ))). Thus �P inducesa homomorphism

�P : AutF(P ) !N�(�(CS(P )))=�(CS(P )):

We now make use of Lemma 4.1, which gives suÆcient conditions to the existence ofan extension of a fusion mapping triple.

If � 2 AutF(P ), and x 2 �P (�), then x = y ��(h) for some h 2 CS(P ) and y 2 �(�),where � is an extension of � to P � CS(P ). Hence for any g 2 P ,

x�(g)x�1 = y � �(hgh�1)y�1 = y�(g)y�1 = �(�(g)) = �(�(g)):

This shows that point (+) of Lemma 4.1 holds, and so it remains to check (�).

Assume P � Q � S, P C Q, and let � 2 AutF(P ) and � 2 AutF(Q) be such that� = �jP . Then Q�CS(P ) � N� in the terminology of axiom (II), so � extends to anotherautomorphism 2 AutF(Q�CS(P )), and �P (�) = �( )��(CS(P )) by de�nition of �P .By [5A1, Lemma 3.8] again, jQ = cg Æ � for some g 2 CS(P ). Hence, by De�nition3.6, �( ) = �(cg Æ �) = �(g) ��(�), and so

�P (�) = �(g) ��(�) � �(CS(P )) = �(�)�(�(g)) � �(CS(P )) = �(�) � �(CS(P )):

In particular, �P (�) � �(�). This shows that point (�) of Lemma 4.1 is satis�ed aswell, and thus, by the lemma, � can be extended to a fusion mapping triple on FH. �


Recall that a fusion subsystem of index prime to p in a saturated fusion system Fover S is a saturated subsystem F0 � F over the same p-group S, such that AutF0(P ) �Op0(AutF(P )) for all P � S. Equivalently, F0 � F has index prime to p if and onlyif it is saturated and contains the subcategory Op0

� (F) of De�nition 3.3. We are nowready to prove our main result about these subsystems.

Theorem 5.4. For any saturated fusion system F over a p-group S, there is a bijectivecorrespondence between subgroups

H � �p0(F) = OutF(S)=Out0F (S);

and saturated fusion subsystems FH of F over S of index prime to p in F . The

correspondence is given by associating to H the fusion system generated by b��1(B(H)),

where b� is the functor of Proposition 5.2.

Proof. Let F0 � F be any saturated fusion subsystem over S which contains Op0

� (F).Then Out0F(S) C OutF0(S), and one can set H = OutF0(S)=Out

0F(S). We �rst show

that a morphism ' of F c is in F0 if and only if b�(') 2 H. Clearly it suÆces to provethis for isomorphisms in F c.

Let P;Q � S be F -centric, F -conjugate subgroups, and �x an isomorphism ' 2IsoF(P;Q). By Lemma 3.4, we can write ' = Æ (�jP ), where � 2 AutF(S) and 2 Iso

Op0

� (F)(�(P ); Q). Then ' is in F0 if and only if �jP is in F0. Also, by de�nition

of b� (and of H), b�(') 2 H if and only if � 2 AutF0(S). So it remains to prove that�jP 2 Mor(F0) if and only if � 2 AutF0(S).

If � 2 AutF0(S), then �jP is also in F0 by de�nition of a fusion system. So itremains to prove the converse. Assume �jP is in F0. The same argument as that usedto prove Proposition 3.8(c) shows that �(P ) is F0-centric, and hence fully centralizedin F0. Since �jP extends to an (abstract) automorphism of S, axiom (II) implies thatit extends to some �1 2 HomF0(NS(P ); S). Since P is F -centric, [BLO2, PropositionA.8] applies to show that �1 = (�jNS(P ))Æcg for some g 2 Z(P ), and thus that �jNS(P ) 2HomF0(NS(P ); S). Also, NS(P ) P whenever P � S, and so we can continue this

process to show that � 2 AutF0(S). This �nishes the proof that F0 = b��1(H).

Now �x a subgroup H � OutF(S)=Out0F (S), and let FH be the smallest fusion

system over S which contains b��1(B(H)). We must show that FH is a saturated fusionsubsystem of index prime to p in F . For F -centric subgroups P;Q � S, HomFH (P;Q)

is the set of all morphisms ' in HomF(P;Q) such that b�(') 2 H. In particular,

FH � Op0

� (F), since morphisms in Op0

� (F) are sent by b� to the identity element.

De�ne �: Mor(F c) !Sub(�p0(F)) by setting �(') = fb�(')g; i.e., the imageconsists of subsets with one element. Let � 2 Hom(S;�p0(F)) be the trivial (andunique) homomorphism. Then (�p0(F); �;�) is a fusion mapping triple on F c. ByLemma 5.3, this can be extended to a fusion mapping triple on F q; and hence FH issaturated by Proposition 3.8.

By Theorem 1.5(a) (Alperin's fusion theorem), FH is the unique saturated fusionsubsystem of F with the property that a morphism ' 2 HomF(P;Q) between F -centric

subgroups of S lies in FH if and only if b�(') 2 H. This shows that the correspondenceis indeed bijective. �

The next theorem describes the relationship between subgroups of index prime to pin a p-local �nite group (S;F ;L) and certain covering spaces of jLj.


Theorem 5.5. Fix a p-local �nite group (S;F ;L). Then for each subgroup H �OutF(S) containing Out0F(S), there is a unique p-local �nite subgroup (S;FH ;LH),such that FH has index prime to p in F , OutFH (S) = H, and LH = ��1(FH) (where� is the usual functor from L to F). Furthermore, jLHj is homotopy equivalent, via

its inclusion into jLj, to the covering space of jLj with fundamental group eH: whereeH � �1(jLj) is the subgroup such that ��( eH=Op0(�1(jLj))) corresponds to H=Out0F(S)under the isomorphism

�1(jLj)=Op0(�1(jLj)) �= �1(jF

cj)=Op0(�1(jFcj))

��!

�=OutF(S)=Out

0F(S)

of Proposition 5.2.

Proof. By Theorem 3.9, applied to the composite functor

L�

��! F cb�

��! B(OutF(S)=Out0F (S));

(S;FH ;LH) is a p-local �nite group, and jLH j is homotopy equivalent to the covering

space of jLj with fundamental group eH de�ned above. The uniqueness follows fromTheorem 5.4. �

Theorem 5.4 shows, in particular, that any saturated fusion system F over S containsa unique minimal saturated subsystem Op0(F) of index prime to p: the subsystemF0 � F with OutF0(S) = Out0F(S). Furthermore, if F has an associated centric linkingsystem L, then Theorem 5.5 shows thatOp0(F) has an associated linking system Op0(L),whose geometric realization jOp0(L)j is homotopy equivalent to the covering space ofjLj with fundamental group Op0(�1(jLj)).

5.2 Extensions of index prime to p It remains to consider the opposite prob-lem: describing the extensions of a given saturated fusion system of index prime to p.As before, for a saturated fusion system F over a p-group S, Autfus(S;F) denotes thegroup of fusion preserving automorphisms of S (De�nition 4.5). Theorem 5.7 belowstates that each subgroup of Outfus(S;F) = Autfus(S;F)=AutF(S) of order prime to pgives rise to an extension of F .

The following lemma will be needed to compare the obstructions to the existence anduniqueness of linking systems, in a fusion system and in a fusion subsystem of indexprime to p. Recall the de�nition of the orbit category of a fusion system in De�nition1.7.

Lemma 5.6. Fix a saturated fusion system F over a p-group S, and let F 0 � F bea saturated subsystem of index prime to p. Assume OutF 0(S) C OutF(S), and set� = OutF(S)=OutF 0(S). Then for any F : Oc(F) ! Z(p)-mod, there is a naturalaction of � on the higher limits of F jOc(F 0), and

lim ��

Oc(F)

(F ) �=hlim �

�

Oc(F 0)

�F jOc(F 0)

�i�:

Proof. Let Oc(F)-mod denote the category of functors Oc(F)op ! Ab. For anyF in Oc(F)-mod, OutF(S) acts on

QP2Ob(Oc(F)) F (P ) by letting � 2 OutF(S) send

F (�(P )) to F (P ) via the induced map ��. This restricts to an action of � on �(F )def=


lim �Oc(F 0)

(F jOc(F 0)); and by de�nition of inverse limits,

lim �Oc(F)

(F ) �=hlim �Oc(F 0)

(F jOc(F 0))i�

=��(F )

��:

Since F 0 is a subsystem of index prime to p, OutF 0(S) is a subgroup of Out0F(S).

Hence, by a slight abuse of notation, one has a functor b� : F ! B(�) given as the

composite of the functor b� of Proposition 5.2 with projection to �. For any Z[�]-module M , regarded as a functor on B(�), we let �(M) denote the composite functor

M Æ b�. Then Z[�]-mod�!Oc(F)-mod is an exact functor, and a left adjoint to

Oc(F)-mod�! Z[�]-mod. In particular, the existence of a left adjoint shows that �

sends injective objects in Oc(F)-mod to injective Z[�]-modules.

Thus, if 0! F ! I0 ! I1 ! � � � is an injective resolution of F in Oc(F)-mod, andF takes values in Z(p)-modules, then

lim ��

Oc(F)

(F ) �= H��0! �(I0)

�(p) ! �(I1)

�(p) ! � � �

�

�=�H�

�0 ! �(I0)(p) ! �(I1)(p) ! � � �

�� = hlim �

�

Oc(F 0)

(F jOc(F 0))i�: �

We are now ready to examine extensions of index prime to p.

Theorem 5.7. Fix a saturated fusion system F over a p-group S. Let

� � Outfus(S;F)def= Autfus(S;F)=AutF(S)

be any subgroup of order prime to p, and let e� denote the inverse image of � inAutfus(S;F).

(a) Let F :� be the fusion system over S generated (as a category) by F together withrestrictions of automorphisms in e�. Then F :� is a saturated fusion system, whichcontains F as a fusion subsystem of index prime to p.

(b) If F has an associated centric linking system, then so does F :�.

(c) Let L be a centric linking system associated to F . Assume that for each � 2 �, theaction of � on F lifts to an action on L. Then there is a unique centric linkingsystem L:� associated to F :� whose restriction to F is L.

Proof. (a) By de�nition, every morphism in F :� is the composite of morphisms in Fand restrictions of automorphisms of S which normalize F . If 2 HomF(P;Q) and' 2 Autfus(S;F), then one has

'jQ Æ = 'jQ Æ Æ '�1j'(P ) Æ 'jP = 0 Æ 'jP ; (1)

where 0 2 HomF('(P ); '(Q)) (since ' is fusion preserving). Hence each morphism inF :� is the composite of the restriction of a morphism in e� followed by a morphism inF .

We next claim F is a fusion subsystem of F :� of index prime to p. More precisely,we will show, for all P � S, that AutF (P ) C AutF :�(P ) with index prime to p. To seethis, let �1 � e� be the set of automorphisms ' in e� � Autfus(S;F) such that '(P ) isF -conjugate to P , and let �0 � �1 be the set of classes ' such that 'jP 2 HomF (P; S).If '(P ) and (P ) are both F -conjugate to P , then ('(P )) is F -conjugate to (P )since is fusion preserving, and thus is F -conjugate to P . This shows that �1 is


a subgroup of e�, and an argument using (1) shows that �0 is also a subgroup. Byde�nition, �0 � AutF(S), so �1=�0 has order prime to p since � does. Using (1), de�ne

�P : AutF :�(P ) ��! �1=�0

by setting �P (�) = [ ] if � = j�(P )Æ� for some � 2 HomF(P; S) and some �AutF (S) 2�. By de�nition of �0, this is well de�ned, and Ker(�P ) = AutF(P ). Since �1=�0 hasorder prime to p, this shows that AutF(P ) C AutF :�(P ) with index prime to p.

We next claim that F and F :� have the same fully centralized, fully normalized andcentric subgroups (compare with the proof of Proposition 3.8(c)). By de�nition, Fand F :� are fusion systems over the same p-group S. Since each F -conjugacy class iscontained in some F :�-conjugacy class, any subgroup P � S which is fully centralized(fully normalized) in F :� is fully centralized (fully normalized) in F . By the sameargument, any F :�-centric subgroup is also F -centric.

Conversely, assume that P is not fully centralized in F :�, and let P 0 � S be asubgroup F :�-conjugate to P such that jCS(P

0)j > jCS(P )j. Let : P0 ! P be an

F :�-isomorphism between them. Then, by the argument above = 0 Æ', where 0 isa morphism in F and ' 2 e� is the restriction to P 0 of an automorphism of S. HencejCS('(P

0))j = jCS(P0)j > jCS(P )j, and since '(P 0) is F -conjugate to P , this shows

that P is not fully centralized in F . A similar argument shows that if P is not fullynormalized (or not centric) in F :�, then it is not fully normalized (or not centric) inF .

We prove that F :� is saturated using Theorem 1.5(b). Thus, we must show thatconditions (I) and (II) of De�nition 1.2 are satis�ed for all F :�-centric subgroups, andthat F :� is generated by restrictions of morphisms between its centric subgroups. LetP � S be a subgroup which is fully normalized in F :�. Then it is fully normalized in F ,and since F is saturated, it is fully centralized there and AutS(P ) 2 Sylp(AutF(P )) =Sylp(AutF :�). Hence, condition (I) holds for any P in (F :�)c.

Let : P !Q be a morphism in F :�, and write = 0 Æ ', as before. Set

N = fg 2 NS(P ) j Æ cg Æ �1 2 AutS( (P ))g;

then '(N ) = N 0 since 'cg'�1 = c'(g) for all g 2 S. Since condition (II) holds for

F , the morphism 0 can be extended to N 0 . Hence = 0 Æ ' can be extended toN = '(N 0), and so condition (II) holds for F :�.

That all morphisms in F :� are composites of restrictions of F :�-morphisms betweenF :�-centric subgroups holds by construction. Thus Theorem 1.5(b)applies, and F :� issaturated.

(b) Let ZF : Oc(F) ! Ab be the functor ZF (P ) = Z(P ), and similarly for ZF :�(see De�nition 1.7). By Lemma 5.6, restriction of categories induces a monomorphism

lim ��

Oc(F :�)

(ZF :�) ��! lim ��

Oc(F)

(ZF) (2)

whose image is the subgroup of �-invariant elements.

By [BLO2, Proposition 3.1], the obstruction �(F :�) to the existence of a centriclinking system associated to F :� lies in lim �

3(ZF :�). From the construction in [BLO2]of these obstructions (and the fact that F :� and F have the same centric subgroups),it is clear that the restriction map (2) sends �(F :�) to �(F). So if there is a linkingsystem L associated to F , then �(F) = 0, so �(F :�) = 0 by Lemma 5.6, and there isa linking system L:� associated to F :�.


(c) Let L:� be a centric linking system associated to F :�, as constructed in (b), andlet L0 be its restriction to F (i.e., the inverse image of F under the the projectionL:� ! (F :�)c). By [BLO2, Proposition 3.1] again, the group lim �

2(ZF ) acts freelyand transitively on the set of all centric linking systems associated to F . If the actionon F of each � 2 � lifts to an action on L, then the element of lim �

2(ZF) which measuresthe di�erence between L and L0 is �-invariant, and hence (by Lemma 5.6 again) is therestriction of an element of lim �

2(ZF :�). Upon modifying the equivalence class of L:�by this element, if necessary, we get a centric linking system associated to F :� whoserestriction to F is L. �

As was done in the last section for extensions with p-group quotient, we now translatethis last result to a theorem stated in terms of �bration sequences.

Theorem 5.8. Fix a p-local �nite group (S;F ;L), a �nite group � of order prime to

p, and a �bration Ev��! B� with �ber X ' jLj^p . Then there is a p-local �nite group

(S;F 0;L0) such that F � F 0 is normal of index prime to p, AutF 0(S)=AutF (S) is aquotient group of �, and E^

p ' jL0j^p .

Proof. For any space Y , let Aut(Y ) denote the topological monoid of homotopy equiv-

alences Y'��! Y . Fibrations with �ber Y and base B are classi�ed by homotopy

classes of maps B ! BAut(Y ). This follows, for example, as a special case of themain theorem in [DKS].

We are thus interested in the classifying space B Aut(jLj^p ), whose homotopy groupswere determined in [BLO2, x8]. To describe these, let Auttyp(L) be the monoid ofisotypical self equivalences of the category L; i.e., the monoid of all equivalences ofcategories 2 Aut(L) such that for all P 2 Ob(L), P;P (Im(ÆP )) = Im(Æ (P )). LetOuttyp(L) be the group of all isotypical self equivalences modulo natural isomorphismsof functors. By [BLO2, Theorem 8.1], �i(B Aut(jLj^p )) is a �nite p-group for i = 2 andvanishes for i > 2, and

�1(B Aut(jLj^p ))�= Outtyp(L):

Each isotypical self equivalence of L is naturally isomorphic to one which sendsinclusions to inclusions (this was shown in [BLO1, Lemma 5.1] for linking systems ofa group, and the general case follows by the same argument). Thus each element ofOuttyp(L) is represented by some � which sends inclusions to inclusions. This in turnimplies that �S | the restriction of �S;S to ÆS(S) � AutL(S) | lies in Autfus(S;F),and that for every F -centric subgroup P of S, the functor � sends P to the subgroup�S(P ) � S. In particular, � is an automorphism of L, since it induces a bijection onthe set of objects.

Now let B�fv! B Aut(jLj^p ) be the map which classi�es the �bration E

v��! B�.

Since � is a p0-group, the �bration Ev��! B� is uniquely determined by the action

of � on jLj^p ; more precisely, by the map induced by fv on fundamental groups:

�1(fv) : � ��! Outtyp(L) :

By an argument identical to that used to prove [BLO1, Theorem 6.2], there is anexact sequence

0 ��! lim �1

Oc(F)

(ZF) ��! Outtyp(L)�L��! Outfus(S;F);


where �L is de�ned by restricting a functor L ! L to ÆS(S) � AutL(S). Also, Oc(F)and ZF are as in De�nition 1.7, but all that we need to know here is that Ker(�L) isa p-group. Set

v = �L Æ �1(fv) : � ��! Outfus(S;F):

Set � = Im( v) � Outfus(S;F). By its de�nition, v comes equipped with a lift toOuttyp(L). So by the above remarks, every element of � lifts to an automorphism ofL. Let (S;F :�;L:�) be the p-local �nite group constructed in Theorem 5.7(a,c) as anextension of (S;F ;L). This induces a �bration

jLj ��! jL:�j ��! B�:

By [BK, Corollary I.8.3], there is a �berwise completion Ew of jL:�j which sits in a�bration sequence

jLj^p ��! Eww

��! B�;

and this �bration is classi�ed by a map

fw : B� ��! B Aut(jLj^p ):

As was the case for fv, the induced map between fundamental groups �1(fw) determinesthe �bration.

Consider the diagram

��

! �

��

�1(fw)

Outtyp(L)

�1(fv)#

�L! Outfus(S;F) ,

incl#

(1)

where � : � ! � is the restriction of v to its image, and where the square commutessince each composite is equal to v. The composite �L Æ�1(fw) 2 Hom(�;Outfus(S;F))is the homomorphism induced by the homotopy action of � = �1(B�) on the �berjLj^p . By the construction in Theorem 5.7, this is just the inclusion map. Hence thelower right triangle in the above diagram commutes. Since � has order prime to p andKer(�L) �= lim �

1(ZF) is a p-group, any homomorphism from � to Outfus(S;F) has aunique conjugacy class of lifting to Outtyp(L) by the Schur-Zassenhaus theorem (cf.[Go, Theorem 6.2.1]). In particular, �1(fw) Æ � and �1(fv) are conjugate as homomor-phisms from � to Outtyp(L). Thus, the upper left triangle in (1) commutes up toconjugacy.

Since �2(B Aut(jLj^p )) is a �nite p-group and the higher homotopy groups all vanish,we have shown that fw Æ B� is homotopic to fv. In other words, we have a map of�brations

jLj^p ! Ev! B�

jLj^p

wwww! Ew

�#

w! B�;

B�#

A comparison of the spectral sequences for these two �brations shows that � is an Fp -homology equivalence. Since jLj is p-good [BLO2, Proposition 1.12], the natural mapfrom jL:�j to its �berwise completion Ew is also an Fp-homology equivalence. Hencethese maps induce homotopy equivalences E^

p ' (Ew)^p ' jL:�j

^p . This �nishes the

proof of the theorem, with F 0 = F :� and L0 = L:�. �


Note that we are not assuming that � injects into Outfus(S;F) in the hypotheses ofTheorem 5.8. Thus, for example, if we start with a product �bration E = jLj^p � B�,then we end up with (S;F 0;L0) = (S;F ;L) (and E^

p ' jLj^p ).

We do not know whether the index prime to p analog of Corollary 4.8 holds. Thekey point in the proof of Corollary 4.8 is the observation that if G is a �nite group,whose fusion system at p contains a normal subsystem of index pm, then G containsa normal subgroup of index pm. This is not true for subsystems of index prime to p.For example, the fusion system at the prime 3 of the simple groups J4 and Ru has asubsystem of index 2.

6. Central extensions of fusion systems and linking systems

Let F be a fusion system over a p-group S. We say that a subgroup A � S iscentral in F if CF(A) = F , where CF(A) is the centralizer fusion system de�ned in[BLO2, De�nition A.3]. Thus A is central in F if A � Z(S), and each morphism ' 2HomF (P;Q) in F extends to a morphism ' 2 HomF(PA;QA) such that 'jA = IdA.

In this section, we �rst study quotients of fusion systems, and of p-local �nite groups,by central subgroups. Afterwards, we will invert this procedure, and study centralextensions of fusion systems and p-local �nite groups.

6.1 Central quotients of fusion and linking systems We �rst note that ev-ery saturated fusion system F contains a unique maximal central subgroup, which weregard as the center of F .

Proposition 6.1. For any saturated fusion system F over a p-group S, de�ne

ZF(S) =�x 2 Z(S)

��'(x) = x; all ' 2 Mor(F c)= lim �

Fc

Z(�) :

the inverse limit of the centers of all F-centric subgroups of S. Then ZF(S) is thecenter of F : it is central in F , and contains all other central subgroups.

Proof. By de�nition, if A is central in F , then A � Z(S), and any morphism in Fbetween subgroups containing A restricts to the identity on A. Since all F -centricsubgroups contain Z(S), this shows that A � ZF(S).

By Alperin's fusion theorem (Theorem 1.5(a)), each morphism in F is a composite ofrestrictions of morphisms between F -centric subgroups. In particular, each morphismis a restriction of a morphism between subgroups containing ZF(S) which is the identityon ZF(S), and thus ZF(S) is central in F . �

The center of a fusion system F has already appeared when studying mapping spacesof classifying spaces associated to F . By [BLO2, Theorem 8.1], for any p-local �nitegroup (S;F ;L),

Map(jLj^p ; jLj^p )Id ' BZF(S):

We next de�ne the quotient of a fusion system by a central subgroup.

De�nition 6.2. Let F be a fusion system over a p-group S, and let A be a centralsubgroup. De�ne F=A to be the fusion system over S=A with morphism sets

HomF=A(P=A;Q=A) = Im�HomF(P;Q) ��! Hom(P=A;Q=A)

�:


By [BLO2, Lemma 5.6], if F is a saturated fusion system over a p-group S, andA � ZF(S), then F=A is also saturated as a fusion system over S=A. We now want tostudy the opposite question: if F=A is saturated, is F also saturated? The followingexample shows that it is very easy to construct counterexamples to this. In fact, undera mild hypothesis on S, if a fusion system over S=A has the form F=A for any fusionsystem F over S, then it has that form for a fusion system F which is not saturated.

Example 6.3. Fix a p-group S and a central subgroup A � Z(S). Assume there is

� 2 Aut(S)rInn(S) such that �jA = IdA and � induces the identity on S=A. Let F bea saturated fusion system over S=A, and assume there is some fusion system F0 over

S such that F0=A = F . Let F � F0 be the fusion system over S de�ned by setting

HomF(P;Q) =�' 2 Hom(P;Q)

��9' 2 Hom(PA;QA);

'jP = '; 'jA = IdA; '=A 2 HomF(PA=A;QA=A)

:

Then A is a central subgroup of F , and F=A = F . But F is not saturated, sinceAutF (S) contains as normal subgroup the group of automorphisms of S which are theidentity on A and on S=A (a p-group by Lemma 1.15), and this subgroup is by assump-tion not contained in Inn(S). Hence AutS(S) is not a Sylow subgroup of AutF(S).

The hypotheses of the above example are satis�ed, for example, by any pair of p-groups 1 6= A � S with A � Z(S), such that S is abelian, or more generally such thatA \ [S; S] = 1.

We will now describe conditions which allow us to say when F is saturated. Beforedoing so, in the next two lemmas, we �rst compare properties of subgroups in F withthose of subgroups of F=A, when F is a fusion system with central subgroup A. Thisis done under varying assumptions as to whether F or F=A is saturated.

Lemma 6.4. The following hold for any fusion system F over a p-group S, and anysubgroup A � Z(S) central in F .

(a) If A � P � S and P=A is F=A-centric, then P is F-centric.

(b) If F=A is saturated, and P � S is F-quasicentric, then PA=A is F=A-quasicentric.

(c) If F and F=A are both saturated and P=A � S=A is F=A-quasicentric, then P isF-quasicentric.

Proof. For each P � S containing A, let eCS(P ) � S be the subgroup such thateCS(P )=A = CS=A(P=A). Let

�P : eCS(P ) ��! Hom(P;A)

be the homomorphism �P (x)(g) = [x; g] for x 2 eCS(P ) and g 2 P . Thus Ker(�P ) =CS(P ).

For each P � S containing A, set

�P = Ker�AutF(P ) ��! AutF=A(P=A)

�:

Since every � 2 AutF(P ) is the identity on A, �P is a p-group by Lemma 1.15.

(a) If g; g0 commute in S, then their images commute in S=A. Thus for all Q � S,CS(Q)=A � CS=A(Q=A).


Assume P=A is F=A-centric. Then for each P 0 which is F -conjugate to P , P 0=Ais F=A-conjugate to P=A, and hence CS(P

0)=A � CS=A(P0=A) � P 0=A. Thus P is

F -centric.

(b) Fix P � S such that P is F -quasicentric. Since CS(P0A) = CS(P

0) for all P 0

which is F -conjugate to P , P 0A is fully centralized in F if and only if P 0 is. Also,CF(P

0) = CF(P0A) for such P 0, and this shows that PA is also F -quasicentric. So

after replacing P by PA, if necessary, we can assume P � A.

Choose P 0=A which is fully centralized in F=A and F=A-conjugate to P=A. ThenP 0 is F -conjugate to P , hence still F -quasicentric. So upon replacing P by P 0, we arereduced to showing that P=A is F=A-quasicentric when P � A, P is F -quasicentric,and P=A is fully centralized.

For any P 0 which is F -conjugate to P , there is a morphism

'=A 2 HomF=A

�(P 0� eCS(P 0))=A; (P � eCS(P ))=A�

which sends P 0=A to P=A (by axiom (II) for the saturated fusion system F=A). Then' restricts to a morphism from CS(P

0) to CS(P ). Thus jCS(P 0)j � jCS(P )j for all P 0

which is F -conjugate to P , and this proves that P is fully centralized in F .

If P=A is not F=A-quasicentric, then by Lemma 1.6 (and since F=A is saturated),there is some Q=A � P=A�CS=A(P=A) containing P=A, and some � 2 AutF(Q), suchthat Id 6= �=A 2 AutF=A(Q=A) has order prime to p and �=A is the identity onP=A. Since � is also the identity on A, Lemma 1.15 implies that �jP = IdP . Set

Q0 = Q \ eCS(P ). Then �(Q0) = Q0, and �P Æ � = �P since � is the identity on P � A.Thus � induces the identity on Q0=CQ0(P ) since Ker(�P ) = CS(P ). Since � has orderprime to p, Lemma 1.15 now implies that �jCQ0(P ) 6= Id. Thus CQ0(P ) � CS(P ) and

�jP �CQ0 (P ) is a nontrivial automorphism in CF(P ) of order prime to p. Since P is fully

centralized in F , this implies (by de�nition) that CF(P ) is not the fusion system ofCS(P ), and hence that P is not F -quasicentric.

(c) Now assume that F and F=A are both saturated, and that P=A � S=A isF=A-quasicentric. If P is not F -quasicentric, and P 0 is F -conjugate to P and fullycentralized in F , then by Lemma 1.6(b), there is some P 0 � Q � P 0�CS(P 0) andsome Id 6= � 2 AutF(Q) such that �jP 0 = IdP 0 and � has order prime to p. ThenQ=A � (P 0=A)�CS=A(P

0=A), �=A 2 AutF=A(Q=A) also has order prime to p, and so�=A 6= Id by Lemma 1.15 again. But by Lemma 1.6(a), this contradicts the assump-tion that P=A is F=A-quasicentric. �

In the next lemma, we compare conditions for being fully normalized in F and inF=A.

Lemma 6.5. The following hold for any fusion system F over a p-group S, and anysubgroup A � Z(S) central in F .

(a) Assume F is saturated. Then for all P;Q � S containing A such that P is fullynormalized in F , if '; '0 2 HomF(P;Q) are such that '=A = '0=A, then '0 = ' Æ cxfor some x 2 NS(P ) such that xA 2 CS=A(P=A).

(b) Assume F=A is saturated, and let P; P 0 � S be F-conjugate subgroups whichcontain A. Then P is fully normalized in F if and only if P=A is fully nor-malized in F=A. Moreover, if P is fully normalized in F , then there is 2HomF(NS(P

0); NS(P )) such that (P 0) = P .


Proof. (a) Assume F is saturated, and �x P;Q � S containing A such that P isfully normalized in F . Let '; '0 2 HomF(P;Q) be such that '=A = '0=A. ThenIm(') = Im('0). Set � = '�1 Æ'0 2 AutF(P ); then '

0 = ' Æ�, and �=A = IdP=A. Thus

� 2 Ker[AutF(P ) ��! AutF=A(P=A)];

which is a normal p-subgroup by Lemma 1.15. Since P is fully normalized, AutS(P ) 2Sylp(AutF(P )) and so � 2 AutS(P ). Thus � 2 cx for some x 2 NS(P ), and xA 2CS=A(P=A) since �=A = IdP=A.

(b) We are now assuming that F=A is saturated. Assume �rst that P=A is fullynormalized in F=A. Then by Lemma 1.3, there is a morphism

'0 2 HomF=A(NS=A(P0=A); NS=A(P=A)) (1)

such that '0(P0=A) = P=A, and this lifts to ' 2 HomF(NS(P

0); NS(P )) such that'(P 0) = P . Therefore jNS(P

0)j � jNS(P )j for any P 0 which is F -conjugate to P andP is fully normalized in F . This also proves that the last statement in (b) holds, oncewe know that P=A is fully normalized.

Assume now that P is fully normalized in F ; we want to show that P=A is fullynormalized in F=A. Fix P 0 which is F -conjugate to P and such that P 0=A is fullynormalized in F=A. By Lemma 1.3 again, there exists

'0 2 HomF=A(NS=A(P=A); NS=A(P0=A))

such that '0(P=A) = P 0=A. Then '0 = '=A for some ' 2 HomF(NS(P ); NS(P0)), and

' is an isomorphism since P is fully normalized in F . Thus '0 is an isomorphism, andhence P=A is fully normalized in F=A.

Thus if P is fully normalized in F , then P=A is also fully normalized, and we havealready seen that the last statement in (b) holds in this case. �

We are now ready to give conditions under which we show that F is saturated ifF=A is saturated. As in [5A1, x2], for any fusion system F over a p-group S, and anyset H of subgroups of S, we say that F is H-generated if each morphism in F is acomposite of restrictions of morphisms between subgroups in H.

Proposition 6.6. Let A be a central subgroup of a fusion system F over a p-groupS, such that F=A is a saturated fusion system. Let H be any set of subgroups of S,closed under F-conjugacy and overgroups, which contains all F-centric subgroups ofS. Assume

(a) Ker�AutF(P ) ��! AutF=A(P=A)

�� AutS(P ) for each P 2 H which is fully nor-

malized in F ; and

(b) F is H-generated.

Then F is saturated.

Proof. Let H0 be the set of all P 2 H such that P � A. Since A is central, eachmorphism ' 2 HomF(P;Q) extends to some ' 2 HomF(PA;QA) such that 'jA = IdA.Thus F isH0-generated if it isH-generated. So upon replacingH byH0, we can assumeall subgroups in H contain A.

By assumption, H contains all F -centric subgroups of S. So by Theorem 1.5(b), itsuÆces to check that axioms (I) and (II) hold for all P 2 H. For all P � S containing


A, we write

�Pdef= Ker

�AutF(P ) ��! AutF=A(P=A)

�:

(I) Assume P 2 H is fully normalized in F . Then P=A is fully normalized in F=A byLemma 6.5(b). By assumption, �P � AutS(P ). Hence

[AutF(P ) : AutS(P )] = [AutF=A(P=A) : AutS=A(P=A)];

and AutS(P ) 2 Sylp(AutF(P )) since AutS=A(P=A) 2 Sylp(AutF=A(P=A)).

Assume P 0 � S is F -conjugate to P and fully centralized in F . By Lemma 6.5(b)again, there is 2 HomF(NS(P

0); NS(P )) such that (P 0) = P . Then (CS(P0)) �

CS(P ), so P is also fully centralized.

(II) Assume ' 2 HomF(P; S) is such that P 2 H and '(P ) is fully centralized. Set

N' = fg 2 NS(P ) j'cg'�1 2 AutS('(P ))g;

as usual. Then N'=A � N'=A.

Assume �rst that '(P ) is fully normalized in F . Then '(P )=A is fully normalizedin F=A by Lemma 6.5(b), and hence also fully centralized. So by (II), applied to thesaturated fusion system F=A, there is b' 2 HomF(N'=A; S=A) such that b'j(P=A) = '=A.Let ' 2 HomF(N'; S) be a lift of b', then ('jP )=A = '=A and '(P ) = '(P ). So thereis � = ' Æ ('jP )�1 2 �'(P ) � AutF('(P )) such that � Æ 'jP = '. By (a), there isx 2 NS('(P )) such that � = cx; then cx Æ ' lies in HomF(N'; S) and extends '.

It remains to prove the general case. Choose P 0 which is fully normalized in F andF -conjugate to P . By Lemma 6.5(b), there is 2 HomF(NS('(P )); NS(P

0)) such that ('(P )) = P 0. Then N' � N '. Since '(P ) = P 0 is fully normalized, ' extends

to some 2 HomF(N'; NS(P0)). We will show that Im( ) � Im( ), and thus there is

' 2 HomF(N'; NS('(P ))) such that 'jP = '.

For each g 2 N', choose x 2 NS('(P )) such that 'cg'�1 = cx; then we obtain

c (x) = cx �1 = c

(g), and so (x) (g)�1 2 CS(P

0). Since '(P ) is fully centralized

in F , (CS('(P ))) = CS(P0), and thus (g) 2 Im( ). �

For example, one can take as the set H in Proposition 6.6 either the set of subgroupsP � S containing A such that P=A is F=A-quasicentric (by Lemma 6.4(b)), or the setof F -centric subgroups of S.

Now let (S;F ;L) be a p-local �nite group. One can also de�ne a centralizer linkingsystem CL(A) when A � S is fully centralized [BLO2, De�nition 2.4]. Since this isalways a linking system associated to CF(A), A is central in L (i.e., CL(A) = L) if andonly if A is central in F . So from now on, by a central subgroup of (S;F ;L), we justmean a subgroup A � Z(S) which is central in F .

De�nition 6.7. Let (S;F ;L) be a p-local �nite group with a central subgroup A. De�neL=A to be the category with objects the subgroups P=A for P 2 Ob(L) (i.e., such thatP is F-centric), and with morphism sets

MorL=A(P=A;Q=A) = MorL(P;Q)=ÆP (A):

Let (L=A)c � L=A be the full subcategory with object set the F=A-centric subgroups inS=A.

Similarly, if Lq is the associated quasicentric linking system, then de�ne Lq=A to bethe category with objects the F=A-quasicentric subgroups of S=A | equivalently, the


subgroups PA=A for P 2 Ob(Lq) | and with morphisms

MorLq=A(P=A;Q=A) = MorLq(P;Q)=ÆP (A):

Note that by Lemma 6.4(a,c), for any P=A � S=A, if P=A is F=A-centric or F=A-quasicentric, then P is F -centric or F -quasicentric, respectively. Thus the categories(L=A)c � L=A and Lq=A are well de�ned.

We are now ready to prove our main theorem about quotient fusion and linkingsystems.

Theorem 6.8. Let A be a central subgroup of a p-local �nite group (S;F ;L) withassociated quasicentric linking system Lq. Let Lq�A � L

q be the full subcategory withobjects those F-quasicentric subgroups of S which contain A. Then the following hold.

(a) (S=A;F=A; (L=A)c) is again a p-local �nite group, and Lq=A is a quasicentric link-ing system associated to F=A.

(b) The inclusions of categories induce homotopy equivalences jLj ' jLq�Aj ' jLqj and

j(L=A)cj ' jL=Aj ' jLq=Aj.

(c) The functor � : Lq !Lq=A, de�ned by �(P ) = PA=A and with the obvious mapson morphisms, induces principal �bration sequences

BA ��! jLjj� j

��! jL=Aj and BA ��! jLq�Ajj� j

��! jLq=Aj

which remain principal �bration sequences after p-completion.

Proof. (a) The �rst statement is shown in [BLO2, Lemma 5.6] when jAj = p, andthe general case follows by iteration. So we need only prove that Lq=A is a quasicen-tric linking system associated to F=A. Axioms (B)q, (C)q, and (D)q for Lq=A followimmediately from those axioms applied to Lq, so it remains only to prove (A)q.

Let H be the set of subgroups P � S such that A � P and P=A is F=A-quasicentricand �x P;Q 2 H. By construction, CS=A(P=A) acts freely on MorLq=A(P=A;Q=A) andinduces a surjection

MorLq=A(P=A;Q=A)ÆCS=A(P=A) �� HomF=A(P=A;Q=A):

We must show that this is a bijection whenever P=A is fully centralized in F=A. Sinceany other fully centralized subgroup in the same F -conjugacy class has centralizerof the same order, it suÆces to show this when P=A is fully normalized in F=A; orequivalently (by Lemma 6.5(b)) when P is fully normalized in F .

Let eCS(P ) � S be the subgroup such that eCS(P )=A = CS=A(P=A). Fix any F=A-quasicentric subgroup Q=A, and consider the following sequence of maps

MorLq(P;Q) ��! HomF(P;Q) ��! HomF=A(P=A;Q=A):

Since Lq is the quasicentric linking system of F , by property (A)q the �rst map isthe orbit map of the action of CS(P ). The second is the orbit map for the ac-tion of Aut

eCS(P )(P ) by Lemma 6.5(a). Thus the composite is the orbit map for

the free action of eCS(P ). It now follows that eCS(P )=A �= CS=A(P=A) acts freely onMorLq=A(P=A;Q=A) = MorLq(P;Q)=A with orbit set HomF=A(P=A;Q=A).

(b) These homotopy equivalences are all special cases of Proposition 1.12(a).

(c) For any category C and any n � 0, let Cn denote the set of n-simplices in thenerve of C; i.e., the set of composable n-tuples of morphisms. For each n � 0, the


group B(A)n acts freely on (Lq�A)n: an element (a1; : : : ; an) acts by composing the i-thcomponent with ÆP (ai) for appropriate P . This action commutes with the face anddegeneracy maps, and its orbit set is (Lq=A)n. It follows that the projection of jLq�Ajonto jLq=Aj (and of jLj onto jL=Aj) is a principal �bration with �ber the topologicalgroup BA = jB(A)j (see, e.g., [May, xx18{20] or [GJ, Corollary V.2.7]).

By the principal �bration lemma of Bous�eld and Kan [BK, II.2.2], these sequencesare still principal �bration sequences after p-completion. �

6.2 Central extensions of p-local �nite groups We �rst make it more precisewhat we mean by this.

De�nition 6.9. A central extension of a (saturated) fusion system F over a p-group

S by an abelian p-group A consists of a (saturated) fusion system eF over a p-group eS,together with an inclusion A � Z(eS), such that A is a central subgroup of eF, eS=A �= S,

and eF=A �= F as fusion systems over S.

Similarly, a central extension of a p-local �nite group (S;F ;L) by an abelian group

A consists of a p-local �nite group (eS; eF ; eL), together with an inclusion A � ZeF(eS),

such that (eS=A; eF=A; ( eL=A)c) �= (S;F ;L).

Extensions of categories were de�ned and studied by Georges Ho� in [Hf], where heproves that they are classi�ed by certain Ext-groups. We will deal with one partic-ular case of this. Ho�'s theorem implies that an extension of categories of the typeB(A) ��! eLq�A �

��! Lq is classi�ed by an element in lim �Lq

2(A). What this extension

really means is that Lq is a quotient category of eLq�A, where each morphism set in eLq�Aadmits a free action of A with orbit set the corresponding morphism set in Lq. Also,lim �

2(A) means the second derived functor of the limit of the constant functor whichsends each object of Lq to A and each morphism to IdA.

We regard A as an additive group. Fix an element [!] 2 lim �Lq

2(A), where ! is a

(reduced) 2-cocyle. Thus ! is a function from pairs of composable morphisms in Lq toA such that !(f; g) = 0 if f or g is an identity morphism, and such that for any triplef; g; h of composable morphisms, the cocycle condition is satis�ed

d!(f; g; h)def= !(g; h)� !(gf; h) + !(f; hg)� !(f; g) = 0:

Consider the composite i Æ ÆS : BS ! Lq, where ÆS is induced by the distinguishedmorphism S ! AutLq(S) and i is induced by the inclusion of AutLq(S) in Lq as thefull subcategory with one object S. Then !S = (i Æ ÆS)

�(!) is a 2-cocycle de�ned on Swhich classi�es a central extension of p-groups

1 ��! A ��! eS ��! S ��! 1 :

Thus eS = S � A, with group multiplication de�ned by

(h; b)�(g; a) = (hg; b+ a + !S(g; h));

and �(g; a) = g. For each F -quasicentric subgroup P � S, set eP = ��1(P ) � eS.Using this cocycle !, we can de�ne a new category eL0 as follows. There is one objecteP = ��1(P ) of eL0 for each object P of Lq. Morphism sets in eL0 are de�ned by setting

MoreL0( eP; eQ) = MorLq(P;Q)� A:


Composition in eL0 is de�ned by

(g; b) Æ (f; a) = (gf; a+ b + !(f; g)):

The associativity of this composition law follows since d! = 0. Furthermore, if wechose another representative !+d� where � is a 1-cochain, the categories obtained areisomorphic. This construction comes together with a projection functor � : eL0 !Lq.

Finally, we de�neÆeS : S ��! Aut

eL0(S)

by setting ÆeS(g; a) = (ÆS(g); a). This is a group homomorphism by de�nition of !S.

Proposition 6.10. Let (S;F ;L) be a p-local �nite group, and let Lq be the associatedquasicentric linking system. Fix an abelian group A and a reduced 2-cocycle ! on Lq

with coeÆcients in A. Let

eL0 ��! Lq and eS �

��! S

be the induced extensions of categories and of groups, with distinguished monomorphismÆeS as de�ned above.

Then there is a unique saturated fusion system eF over eS and a functor e� : eL0 ! eF ,together with distinguished monomorphisms Æ

eP :eP ! Aut

eL0(P ).

If eL � eL0 denotes the full subcategory whose objects are the eF-centric subgroups ofeS then (eS; eF ; eL) is a p-local �nite group with central subgroup A � Z(eS), such that

(eS=A; eF=A; eL0=A) �= (S;F ;Lq): (1)

Also, eL0 extends to a quasicentric linking system eLq associated to eF.Proof. Assume that inclusion morphisms �P have been chosen in the quasicentric linkingsystem Lq associated to (S;F ;L). For each F -quasicentric P � S, de�ne

�eP = (�P ; 0) 2 Mor

eL0( eP; eS):

Next, de�ne the distinguished monomorphism

ÆeP :

eP �CeS(eP ) ��! Aut

eL0( eP )

to be the unique monomorphism such that the following square commutes for each ePand each q 2 eP �C

eS(eP ):

eP �eP ! eS

ePÆeP(q;a)#

�eP ! eS .

ÆeS(q;a)# (2)

More precisely, since ÆeS(q; a) = (ÆS(q); a) by de�nition, this means that

ÆeP (q; a) = (ÆP (q); a+ !(�P ; ÆS(q))� !(ÆP (q); �P )):

For each morphism (f; a) 2 MoreL0( eP ; eQ), and each element (q; b) 2 eP , there is a

unique element c 2 A such that the following square commutes:

eP (f;a)! eQ

ePÆeP(q;b)#

(f;a)! eQ .

ÆeQ(�(f)(q);c)# (3)


In this situation, we set

e�(f; a)(q; b) = (�(f)(q); c) 2 eQ:By juxtaposing squares of the form (3), we see that e�(f; a) 2 Hom( eP; eQ), and that thisde�nes a functor e� from eL0 to the category of subgroups of eS with monomorphisms.

De�ne eF to be the fusion system over eS generated by the image of e� and restrictions.By construction, the surjection � : eS ��! S induces a functor �� : eF ��! F betweenthe fusion systems, which is surjective since F is generated by restrictions of morphismsbetween F -quasicentric subgroups (Theorem 1.5(a)). So we can identify F with eF=A.By Lemma 6.4(b), for each eF -quasicentric subgroup P � eS, PA=A is F -quasicentric.

So we can extend eL0 to a category eLq de�ned on all eF-quasicentric subgroups, bysetting

MorLq(P;Q) =�f 2 Mor

eL0(PA;QA)

�� e�(f)(P ) � Q

and ÆP = ÆPA

for each pair P;Q of eF-quasicentric subgroups. We extend e� to eLq in the obvious way.It remains to prove that eF is saturated, and that eLq is a quasicentric linking system

associated to eF . In this process of doing this, we will also prove the isomorphism (1).

Let H be the set of subgroups eP = ��1(P ) � eS for all F -quasicentric subgroupsP � S.

eF is saturated: We want to apply Proposition 6.6 to prove that eF is saturated. ByLemma 6.4(b), H contains all eF-quasicentric subgroups of eS which contain A, and in

particular, all eF-centric subgroups (since every eF -centric subgroup of eS must contain

A). Since eF is H-generated by construction, it remains only to prove condition (a) inProposition 6.6.

Fix some eP = ��1(P ) in H, and let ' 2 AuteF (eP ) be such that ��(') = IdP . Choose

(f; a) 2 e��1('); thus' = e�(f; a) 2 Ker

�Aut

eF(eP ) ��! AutF (P )

�:

Then �(f) = IdP , so f = ÆP (q) for some q 2 CS(P ), and (f; a) = ÆeP (q; c) for some

c 2 A. Since ÆeP is a homomorphism, the de�nition of ' = e�(f; a) via (3) shows that

' = e�(f; a) = eÆeP (q; c) = c(q;c), and thus that ' 2 Aut

eS(eP ). Thus condition (a) in

Proposition 6.6 holds, and this �nishes the proof that eF is saturated.

eLq is a quasicentric linking system associated to eF : The distinguished monomor-phisms Æ

eP , foreP 2 Ob( eLq), were chosen so as to satisfy (D)q, and this was independent

of the choice of inclusion morphisms which lift the chosen inclusion morphisms in Lq.Once the ÆP were determined, then e� was de�ned to satisfy (C)q, and eF was de�nedas the category generated by Im(e�) and restrictions. Thus all of these structures wereuniquely determined by the starting data. Axiom (B)q follows from (C)q by Lemma1.10.

It remains only to prove (A)q. We have already seen that the functor e� is surjective

on all morphism sets. Also, since CeS(P ) = C

eS(PA) for all P �eS, it suÆces to prove

(A)q for morphisms between subgroups eP = ��1(P ) and eQ = ��1(Q) containing A.

By construction, CeS(eP ) acts freely on each morphism set Mor

eLq(eP; eQ), and it remains

to show that if eP is fully centralized, then e�eP ; eQ is the orbit map of this action. As in

the proof of Theorem 6.8(a), it suÆces to do this when eP and P are fully normalized.


Fix two morphisms (f; a) and (g; b) from eP to eQ such that e�(f; a) = e�(g; b). Then�(f) = �(g), so g = f Æ ÆP (x) for some x 2 CS(P ), and

(g; b) = (f; a) Æ (ÆP (x); b� a� !(ÆP (x); f)) = (f; a) Æ ÆeP (x; c);

where c = b � a � !(ÆP (x); f) � (!(�; ÆS(x)) � !(ÆS(x); �P )). Also, e�(ÆeP (x; c)) = Id

eP

implies (via (3)) that (x; c) commutes with all elements of eP , so (x; c) 2 CeS(eP ), and

this �nishes the argument. �

We now want to relate the obstruction theory for central extensions of linking systemswith kernel A to those for central extensions of p-groups, and to those for principal�brations with �ber BA. As a consequence of this, we will show (in Theorem 6.13) thatwhen appropriate restrictions are added, these three types of extensions are equivalent.

Given a central extension eF of F by the central subgroup A, there is an inducedcentral extension 1! A! eS ! S ! 1 of Sylow subgroups. Restriction to subgroupsP � S produces corresponding central extensions 1 ! A ! eP ! P ! 1. Thehomology classes of these central extensions are all compatible with morphisms fromthe fusion system, and hence de�ne an element in lim �

F

H2(�;A). This, together with

notation already used in [BLO2], motivates the following de�nition:

De�nition 6.11. For any saturated fusion system over a p-group S, and any �niteabelian p-group A, de�ne

H�(F ;A) = lim �F

H�(�;A) �= lim �Fc

H�(�;A):

The following lemma describes the relation between the cohomology of F and thecohomology of the geometric realization of any linking system associated to F .

Lemma 6.12. For any p-local �nite group (S;F ;L), and any �nite abelian p-group A,the natural homomorphism

H�(jLj;A) ��! H�(F ;A); (1)

induced by the inclusion of BS into jLj, is an isomorphism. Furthermore, there arenatural isomorphisms

lim �Lq

2(A) �= H2(jLqj;A) �= H2(jLj;A): (2)

Proof. The second isomorphism in (2) follows from Proposition 1.12(a). The �rstisomorphism holds for higher limits of any constant functor over any small, discretecategory C, since both groups are cohomology groups of the same cochain complex

0 ��!Yc

A ��!Yc0!c1

A ��!Y

c0!c1!c2

A ��! � � � :

This cochain complex for higher limits is shown in [GZ, Appendix II, Proposition 3.3](applied withM = Ab

op).

To prove the isomorphism (1), it suÆces to consider the case where A = Z=pn forsome n. This was shown in [BLO2, Theorem 5.8] when A = Z=p, so we can assumethat n � 2, and that the lemma holds when A = Z=pn�1. Consider the following


diagram of Bockstein exact sequences

! H i+1(jLj;Z=p) !H i(jLj;Z=pn�1) !H i(jLj;Z=pn) !H i(jLj;Z=p) !

! H i+1(BS;Z=p)#

!H i(BS;Z=pn�1)#

!H i(BS;Z=pn)#

!H i(BS;Z=p)#

! .

(1)

We claim that the bottom row restricts to an exact sequence of groups H�(F ;�). Oncethis is shown, the result follows by the 5-lemma.

By [BLO2, Proposition 5.5], there is a certain (S; S)-biset which induces, via a sumof composites of transfer maps and maps induced by homomorphisms, an idempotentendomorphism of H�(BS;Z=p) whose image is H�(F ;Z=p). This biset also inducesendomorphisms [] of H�(BS;Z=pn) and H�(BS;Z=pn�1) which commute with thebottom row in (1), since any exact sequence induced by a short exact sequence of coef-�cient groups will commute with transfer maps and maps induced by homomorphisms.The same argument as that used in the proof of [BLO2, Proposition 5.5] shows thatin all of these cases, Im([]) = H�(F ;�), and the restriction of [] to its image ismultiplication by jj=jSj 2 1 + pZ. Thus the sequence of the H�(F ;�) splits as adirect summand of the bottom row in (1), and hence is exact. �

We can now collect the results about central extensions of (S;F ;L) in the followingtheorem, which is the analog for p-local �nite groups of the classical classi�cation ofcentral extensions of groups.

Theorem 6.13. Let (S;F ;L) be a p-local �nite group. For each �nite abelian p-groupA, the following three sets are in one-to-one correspondence:

(a) equivalence classes of central extensions of (S;F ;L) by A;

(b) equivalence classes of principal �brations BA !X ! jLj^p ; and

(c) isomorphism classes of central extensions 1 ! A ! eS �! S ! 1 for

which each morphism ' 2 HomF(P;Q) lifts to some e' 2 Hom(��1(P ); ��1(Q)).

The equivalence between the �rst two is induced by taking classifying spaces, and theequivalence between (a) and (c) is induced by restriction to the underlying p-group.These sets are all in natural one-to-one correspondence with

lim �L

2(A) �= H2(jLj;A) �= H2(F ;A): (1)

Proof. The three groups in (1) are isomorphic by Lemma 6.12. By Proposition 6.10,central extensions of L are classi�ed by lim �

L

2(A). Principal �brations over jLj^p with

�ber BA are classi�ed by�jLj^p ; B(BA)

��= H2(jLj^p ;A)

�= H2(jLj;A);

where the second isomorphism holds since jLj is p-good ([BLO2, Proposition 1.12])and A is an abelian p-group. Central extensions of S by A are classi�ed by H2(S;A),and the central extension satis�es the condition in (c) if and only if the correspondingelement of H2(S;A) extends to an element in the inverse limit H2(F ;A).

A central extension of p-local �nite groups induces a principal �bration of classifyingspaces by Theorem 6.8(c), and this principal �bration restricts to the principal �brationover BS of classifying spaces of p-groups. Thus, if the �bration over jLj is classi�ed


by � 2 H2(jLj;A), then the �bration over BS is classi�ed by the restriction of � toH2(BS;A). It is well known that the invariant in H2(S;A) = H2(BS;A) for a centralextension of S by A is the same as that which describes the principal �bration overBS with �ber BA (see [AM, Lemma IV.1.12]). Since H2(jLj;A) injects into H2(F ;A)(Lemma 6.12), this shows that � is also the class of the 2-cocycle which de�nes theextension of categories. So the map between the sets in (a) and (b) de�ned by takinggeometric realization is equal to the bijection de�ned by the obstruction theory.

Since the isomorphism lim �L

2(A) �= H2(F ;A) is de�ned by restriction to S (as a group

of automorphisms in L), the bijection between (a) and (c) induced by the bijection ofobstruction groups is the same as that induced by restriction to S. �

The following corollary shows that all minimal examples of \exotic" fusion systemshave trivial center.

Corollary 6.14. Let F be a saturated fusion system over a p-group S, and assumethere is a nontrivial subgroup 1 6= A � Z(S) which is central in F . Then F is thefusion system of some �nite group if and only if F=A is.

Proof. Assume F is the fusion system of the �nite group G, with S 2 Sylp(G). Byassumption (A is central in F), each morphism in F extends to a morphism betweensubgroups containing A which is the identity on A. Hence F is also the fusion systemof CG(A) over S, and so F=A is the fusion system of CG(A)=A.

It remains to prove the converse. Assume F=A is isomorphic to the fusion system

of the �nite group G, and identify S=A with a Sylow p-subgroup of G. Since by

Lemma 6.12, H2(F=A;A) �= H2(BG;A) and A is F -central, the cocycle classifying the

extension is in H2(BG;A), and hence there is an extension of �nite groups

1 ��! A ��! G�

��! G ��! 1

with the same obstruction invariant as the extension F !F=A. In particular, we

can identify S = ��1(S=A) 2 Sylp(G), and G = G=A.

We will prove the following two statements:

(a) F has an associated centric linking system L; and

(b) LcS=A(G=A) is the unique centric linking system associated to F=A = FS=A(G=A).

Once these have been shown, then they imply that

(S=A;F=A; (L=A)c) �= (S=A;FS=A(G=A);LcS=A(G=A))

as p-local �nite groups. Hence (S;F ;L) �= (S;FS(G);LcS(G)) as p-local �nite groupsby Theorem 6.13, and thus F is the fusion system of G.

It remains to prove (a) and (b). Let

ZF : Oc(F) ��! Z(p)-mod and ZG : O

cS(G) ��! Z(p)-mod

be the categories and functors of De�nition 1.7. By [O1, Lemma 2.1], ZG can also beregarded as a functor on Oc(FS(G)), and

(c) lim ��

OcS(G)

(ZG) �= lim ��

Oc(FS(G))

(ZG).


By [BLO2, Proposition 3.1], the existence and uniqueness of a centric linking systemdepends on the vanishing of certain obstruction classes: the obstruction to existencelies in lim �

3

OcS(G)

(ZF) and the obstruction to uniqueness in lim �2

OcS(G)

(ZF). Thus (b) follows

from [BLO2, Proposition 3.1] and (c), once we know that lim �2(ZG=A) = 0; and this is

shown in [O1, Theorem A] (if p is odd) or [O2, Theorem A] (if p = 2).

It remains to prove point (a), and we will do this by showing that

lim �3

Oc(F)

(ZF) �= lim �3

OcS(G)

(ZG) = 0: (1)

The last equality follows from [O1, Theorem A] or [O2, Theorem A] again, so it remainsonly to prove the isomorphism.

Let H be the set of subgroups P � S containing A such that P=A is F=A-centric;or equivalently, p-centric in G=A. Let OH(F) � Oc(F) and OH(FS(G)) � Oc(FS(G))be the full subcategories with object sets H.

We claim that

lim ��

OH(F)

(ZF) �= lim ��

Oc(F)

(ZF ) and lim ��

OH(FS(G))

(ZG) �= lim ��

Oc(FS(G))

(ZG) : (2)

If P � S is F -centric but not inH, then there is x 2 NS(P )rP such that cx 2 AutF(P )is the identity on P=A and on A, but is not in Inn(P ). Thus 1 6= [cx] 2 Op(OutF(P )),and so P is not F -radical. By [BLO2, Proposition 3.2], if F : Oc(F)op ! Z(p)-modis any functor which vanishes except on the conjugacy class of P , then lim �

�(F ) �=��(OutF(P );F (P )), where �� is a certain functor de�ned in [JMO, x5]. By [JMO,Proposition 6.1(ii)], ��(OutF(P );F (P )) = 0 for any F , since Op(OutF(P )) 6= 1. Fromthe long exact sequences of higher limits induced by extension of functors, it nowfollows that lim �

�

Oc(F)

(F ) = 0 for any functor F on Oc(F) which vanishes on OH(F); and

this proves the �rst isomorphism in (2). The second isomorphism follows by a similarargument.

The natural surjection of F onto F=A induces isomorphisms of categories

OH(F) �= Oc(F=A) �= OH(FS(G)) : (3)

By the de�nition of H, we clearly have bijections between the sets of objects in thesecategories, so it remains only to compare morphism sets. The result follows fromLemma 6.5(a) for sets of morphisms between pairs of fully normalized subgroups, andthe general case follows since every object is isomorphic to one which is fully normalized.

We next claim that for all i > 0,

lim �i

OH(F)

(ZF ) �= lim �i

Oc(F=A)

(ZF=A) and lim �i

OH(FS(G))

(ZG) �= lim �i

Oc(F=A)

(ZG=A) : (4)

To show this, by (3), together with the long exact sequence of higher limits induced bythe short exact sequence of functors

1 ! A ��! ZF ��! ZF=A ! 1;

we need only show that lim �i

Oc(F=A)

(A) = 0 for all i > 0. Here, A denotes the constant

functor on Oc(F=A) which sends all objects to A and all morphisms to IdA. For eachP 2 H, let FA;P be the functor on Oc(F=A) where FA;P (P 0=A) = A if P 0 is F -conjugateto P and FA;P (P

0=A) = 0 otherwise; and which sends isomorphisms between subgroupsconjugate to P to IdA. By [BLO2, Proposition 3.2], lim �

�(FA;P ) �= ��(OutF=A(P=A);A).


Also, �i(OutF=A(P=A);A) = 0 for i > 0 by [JMO, Proposition 6.1(i,ii)], since the actionof OutF=A(P=A) on A is trivial. From the long exact sequences of higher limits induced

by extension of functors, we now see that lim �i(A) = 0 for all i > 0.

Finally, we claim that

ZF=A �= ZG=A (5)

as functors on Oc(F=A) under the identi�cations in (3). To see this, note that foreach P , (ZF=A)(P ) = Z(P )=A = (ZG=A)(P ); and since these are both identi�ed assubgroups of P=A, any morphism in Oc(F=A) from P=A to Q=A induces the same map(under the two functors) from Z(Q)=A to Z(P )=A. (This argument does not applyto prove that ZF �= ZG. These two functors send each object to the same subgroupof S, but we do not know that they send each morphism to the same homomorphismbetween the subgroups.)

Thus by (2), (4), and (5), for all i > 0,

lim �i

Oc(F)

(ZF) �= lim �i

OH(F)

(ZF ) �= lim �i

Oc(F=A)

(ZF=A)

�= lim �i

Oc(F=A)

(ZG=A) �= lim �i

OH(FS(G))

(ZG) �= lim �i

OcS(G)

(ZG):

This �nishes the proof of (1), and hence �nishes the proof of the corollary. �

References

[AM] A. Adem & J. Milgram, Cohomology of �nite groups, Springer-Verlag (1994)[Al] J. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222{241[BK] P. Bous�eld & D. Kan, Homotopy limits, completions, and localizations, Lecture notes in

math. 304, Springer-Verlag (1972)[5A1] C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver, Subgroup families controlling

p-local �nite groups, Proc. London Math. Soc. (to appear)[BLO1] C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed classifying spaces of

�nite groups, Invent. math. 151 (2003), 611{664[BLO2] C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc.

16 (2003), 779{856[DKS] W. Dwyer, D. Kan, & J. Smith, Towers of �brations and homotopical wreath products, J.

Pure Appl. Algebra 56 (1989), 9{28[GZ] P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag (1967)[Gl] G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403{420[GJ] P. Goerss & R. Jardine, Simplicial homotopy theory, Birkh�auser (1999)[Go] D. Gorenstein, Finite groups, Harper & Row (1968)[Hf] G. Ho�, Cohomologies et extensions de categories, Math. Scand. 74 (1994), 191{207.[JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classi�cation of self-maps of BG via G-

actions, Annals of Math. 135 (1992), 184{270[May] P. May, Simplicial objects in algebraic topology, Van Nostrand (1967)[O1] B. Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Camb.

Phil. Soc. 137 (2004), 321{347[O2] B. Oliver, Equivalences of classifying spaces completed at the prime two, Amer. Math. Soc.

Memoirs (to appear)[Pu2] Ll. Puig, Unpublished notes (ca. 1990)[Pu3] Ll. Puig, The hyperfocal subalgebra of a block, Invent. math. 141 (2000), 365{397[Pu4] L. Puig, Full Frobenius systems and their localizing categories, preprint[St] R. Stancu, Equivalent de�nitions of fusion systems, preprint[Suz2] M. Suzuki, Group theory II, Springer-Verlag (1986)


Departament de Matem�atiques, Universitat Aut�onoma de Barcelona, E{08193 Bel-

laterra, Spain

E-mail address : [email protected]

Departament de Matem�atiques, Universitat Aut�onoma de Barcelona, E{08193 Bel-

laterra, Spain


Department of Mathematics, University of Chicago, Chicago, IL 60637, USA


Department of Mathematical Sciences, University of Aberdeen, Meston Building

339, Aberdeen AB24 3UE, U.K.


LAGA, Institut Galil�ee, Av. J-B Cl�ement, 93430 Villetaneuse, France


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