Simple groups acting on translation planes

Journal of Geometry Voi.29 (1987)

0047-2468/87/020126-1451.50+0.20/0 (c) 1987 Birkh~user Verlag, Basel

SIMPLE GROUPS ACTING ON TRANSLATION PLANES

John Fink and Michael J. Kallaher

We discuss the p o s s i b i l i t y of f i n i t e simple groups acting as co l l ineat ion groups on f i n i t e t ranslat ion planes of odd order with special at tent ion paid to the sporadic simple groups. We assume such a group acts i r reduc ib ly ( in the vector space sense) on the plane. I t is shown that i f the character is t ic of the plane does not div ide the order of the group, then the group cannot be one of eleven sporadic simple groups. Also, i f one of the Mathieu groups acts i r reduc ib ly on a f i n i t e t ranslat ion plane then i t is e i ther Mll or M23.

1. INTRODUCTION

One of the oldest problems in the theory of project ive planes is: Can every

group act as a group of co l l ineat ions on a project ive plane? Mendelsohn [16]

showed that the answer is "yes" i f there is no res t r i c t i on on the order of

the plane; that is , he showed that given a group G there exists an i n f i n i t e

project ive plane with G as a group of co l l ineat ions. However the answer is

not even known when the question is rest r ic ted to the class of f i n i t e trans-

la t ion planes (Liebler [ i i ] ) . In par t i cu la r , l i t t l e is known about the

abstract groups which can occur as co l l ineat ion groups of f i n i t e t ranslat ion

planes with dimension d > 3. This problem is addressed by the present

a r t i c l e .

We study the interp lay between res t r i c t ions on the dimension d of a f i n i t e

t ranslat ion plane, elementary abelian 2-groups acting on the plane, and

i r r e d u c i b i l i t y of a co l l ineat ion group on the underlying vector space. (We

are not using i r r e d u c i b i l i t y in Hering's sense. The t ranslat ion complement

is always reducible in Hering's sense. See Hering [7 ] . ) We do th is in par-

t i cu la r for the sporadic simple groups, especial ly the Mathieu groups. Our

resul ts include the conclusion that some of the Mathieu groups cannot act on

f i n i t e t ranslat ion planes of odd character is t ic without acting reducibly on the underlying vector space. See Theorems 4.3 and 4.4.

Fink and Kallaher 127

I t is reasonable to s t a r t with the simple groups, since every group can be

b u i l t up using simple groups. Every solvable simple groupmthat i s , every

cyc l i c group of prime o rde r - - i s the group of co l l i nea t ions of a f i n i t e t rans-

l a t i on plane. Can every f i n i t e nonsolvable simple group act as a group of

co l l i nea t i ons of a f i n i t e t r ans la t i on plane? Some c e r t a i n l y do. The simple

group SL(2,2 s) is a co l l i nea t i on group of both the Desarguesian plane of order

2 s and the Hall plane of order 2 2s. The Susuki group Sz(2 s) is a co l l i nea t i on

group of the LUneburg plane of order 2 2s and of LUneburg planes of la rger

order. The SL(3,2) = PSL(2,7) is a co l l i nea t i on group of the Lor imer -Rah i l l y

andJohnson-Walker planes of order 16.

Two i n te res t i ng facts stand out. F i r s t , these examples involve t rans la t i on

planes of cha rac te r i s t i c two. No known f i n i t e t rans la t i on plane of odd char-

a c t e r i s t i c has a nonsolvable simple group of co l l i nea t ions . Second, every

nonsolvable simple group which is known to act as a co l l i nea t i on group of a

f i n i t e t r ans la t i on plane also has an i r reduc ib le act ion as a subgroup of the

l i nea r t r ans la t i on complement of some f i n i t e t rans la t i on plane, not necessar-

i l y the same one.

Mason [14] has considered the problem of a nonsolvable Chevalley group G of

cha rac te r i s t i c two act ing i r r educ ib l y on a f i n i t e t rans la t i on plane of char-

a c t e r i s t i c two. He makes the strong assumption that the f i e l d GF(q) over

which G is defined is contained in the kernel of the plane. Under the addi-

t iona l assumption that G is untwisted with a root system having a s ingle root

length he proves that G = PSL(2,q). However, he does not determine the plane

~~

This a r t i c l e begins by studying the general s i tua t ion of a ( f i n i t e ) nonsolv-

able simple group G act ing as a group of co l l i nea t ions on a f i n i t e t rans la t i on

plane ~ of dimension d over i t s kernel K = GF(q), where q = pk with p an odd

prime and k ~ 1. I t is eas i l y proven that the l i nea r t r ans la t i on complement

LTC(~) of ~ contains a subgroup isomorphic to G. Thus, wi thout loss of gener-

a l i t y we may always assume G ~ LTC(~). We prove d = 4m fo r some integer m~ I ,

the invo lu t ions of G are Baer invo lu t ions on ~, and the order of the la rgest

elementary abel ian 2-subgroup must d iv ide d. (See Section 3.)

In Section 4 the assumption is made that the group G acts i r r educ ib l y on the

t rans la t i on plane ~ as a vector space over i t s kernel K; however, we do not

128 Fink and Kallaher

always assume that p and IGI are r e l a t i v e l y prime. Whether or not p and IGI

are r e l a t i v e l y prime, there is a f i n i t e f i e l d extension L of K such that ~ is

a vector space over L of (vector space) dimension 2ds - I , where L has degree s

over K, and G is an absolute ly i r reduc ib le group of l i nea r t ransformations on

V over L. General iz ing the resu l ts of Section 3 we show that the order of the

la rges t elementary abel ian 2-subgroup of G must d iv ide 2ds -1, and that every

invo lu t ion of G must have trace 0 on ~ as a vector space over L.

I f the group G has elementary abelian Sylow 2-subgroups, then G, as a subgroup

of GL(2ds-I,L), is not isomorphic to an irreducible subgroup of GL(2ds-l,r

where ~ is the f ie ld of complex numbers. In other words, the representation

of G on ~ as a vector space over L cannot be ordinary. This result applies to

the groups J l ' PSL(2) with w > 3 and w e • 8), SL(2,2 n) with n ~ 2, and

the groups of Ree type. In the particular case where G = J1 we show that -1 p = 11 and 2ds = 64.

We turn our attention to the sporadic simple groups. First, we assume p# IGI.

Hence, the representation of G on ~ over L must be an ordinary representation.

By then studying the character table of each sporadic simple group eleven

possibi l i t ies for G are eliminated: J1' J2' M12' M22' M24' HS, Sz, Ru, .3,

.1, F23; also, for three possibilit ies--F22, F24, F2--no information was

obtained since we could not find the character table. For each of the remain-

ing twelve possibi l i t ies Table 1 in Section 4 gives the possible values for

2ds -1. We wish to thank R. L. Griess for information on the character tables

of several sporadic groups.

The last half of Section 4 considers the five Mathieu groups. Regardless of

whether or not the characteristic p of ~ divides IGI we prove that the only

possibi l i t ies for G among the five Mathieu groups are Mll and M23; further-

more, i f G is either one of these two groups then its representation must

arise from an ordinary representation over C with 2ds "1 = 16 i f G = M11 and -1

2ds = 896 i f G = M23.

Some comments on the methods used are in order. We attempted to hold assump-

tions about the nature of the group G or about i ts representation on ~ to a

minimum. Stronger assumptions about G or i ts representation might lead to

stronger resu l ts a la Mason [14].


During the research on which th i s a r t i c l e is based and the preparat ion of t h i s

a r t i c l e the authors did not have access to the character tables of several of

the sporadic simple groups. R. L. Griess provided informat ion about most of

these groups, and based upon tha t informat ion Table 1 was determined. We

thank R. L. Griess fo r t h i s help. When the At las of Conway and Newton appears,

i t can be used to complete Table I .

We assume the reader is f a m i l i a r with the basic concepts and resu l t s of group

representat ion theory as given, fo r example in Serre [19]. Also, the reader

should be f a m i l i a r wi th the basic resu l ts of t r ans la t i on planes as given in

Kal laher [10] and LUneburg [12].

We wish to thank T. G. Ostrom fo r many helpfu l conversations and fo r a c r i t i -

cal reading of the manuscript. The resu l t s of Section 2 arose in conversa-

t ions invo lv ing the authors and Professor Ostrom, and he should receive much

of the c red i t f o r them.

2. PRELIMINARY RESULTS ON REPRESENTATIONS

In th i s sect ion two resu l t s concerning representat ions of f i n i t e groups are

given. These w i l l be used in subsequent sect ions. The f i r s t r esu l t deals

with i r r educ ib le representat ions.

THEOREM 2.1: Let G be a group of l i nea r t ransformat ions act ing on a f i n i t e -

dimensional vector space V over the f i n i t e f i e l d K, and assume G acts i r reduc-

i b l y on V. There ex is ts a f i n i t e f i e l d extension L of K such that the fo l low-

ing statements hold:

( i ) V is a vector space over L.

( i i ) The group G is a group of l i nea r t ransformat ions on V as a vector

space over L, and G is absolu te ly i r r educ ib le on th is vector space.

( i i i ) dimE(V ) = dimK(V)/dimk(L ).

PROOF: Let A = K[G], the group algebra of G over K; then V is an i r r educ ib le

A-module. I f L = HomA(V,V) then by Schur's Lemma L is a f i e l d . (See Lemma

2.1 of Dornhoff [ i ] . ) Note that L contains K as a sub f ie ld . Statement ( i )

is c lear .

The m u l t i p l i c a t i o n group L* is j us t the c e n t r a l i z e r of G in GL(V,K). Thus


V is a vector space over L. I f B = L[G], the group algebra of G over L, then

V is an i r r e d u c i b l e L G -module since K ~ L. Thus L' = HomB(V,V) is a f i e l d

by Schur 's Lemma. Furthermore, i f a ~ L' and o ~ O, then TO = aT fo r a l l

T E G and o E HomA(V,) since K ~ L. But then a E L. Thus L' = L. By Theorem

4.2 of Dornhoff [ I ] the group G is abso lu te ly i r r e d u c i b l e on V as a vector

space over L. Statement ( i i i ) fo l lows e a s i l y by a count ing argument.

The second r e s u l t deals w i th completely reduc ib le representat ions.

COROLLARY 2.1 .1 : Let G be a group of l i n e a r t ransformat ions act ing on a

vector space V of dimension m over a f i e l d K, and assume the c h a r a c t e r i s t i c of

K does not d i v ide the order o f G. There ex i s t s subspaces U 1, U 2 . . . . . U e of V

and f i n i t e f i e l d ex ten t ions L I , L 2 . . . . . L e of K such tha t the f o l l ow ing

statements hold:

( i ) V = U I m U 2 m . . . �9 U e, and each U i is a minimal G - i nva r i an t subspace

of V on which G acts i r r e d u c i b l y .

( i i ) For each i = I . . . . . e the subspace U i is a vector space over the

f i e l d L i , and the group G induces on the Li-space U i a subgroup

G-i o f GL(U i , L 1) which acts abso lu te ly i r r e d u c i b l y On U i (as a

vector space over L i ) .

( i i i ) For each i = 1 . . . . . e

d imLi (Ui ) : d imK(Ui) /d imK(Li ) .

PROOF: Since the c h a r a c t e r i s t i c of K does not d iv ide the order of G the group

G acts completely reduc ib ly on V. (Theorem 3.1 of Dornhoff [ I ] . ) Thus the

subspaces U I . . . . . U e e x i s t s a t i s f y i n g statement ( i ) . The existence of the

f i e l d s L i and the v a l i d i t y o f statements ( i i ) and ( i i i ) f o l l ow immediately

from Theorem 2.1.

We want to close t h i s sect ion w i th an extension of a r e s u l t of Ostrom [18]

concerning the r e l a t i o n s h i p between the dimension of a t r a n s l a t i o n plane and

the order of an elementary abel ian group of Baer i n v o l u t i o n s . The dimension

is def ined as fo l l ows .

DEFINITION. Let ~ be a f i n i t e t r a n s l a t i o n plane wi th kernel K = GF(q).

order of ~ is then qd fo r some in teger d ~ i . The dimension of ~ is d.

The

The terminology "dimension" is due to the f o l l ow ing . By a well-known r e s u l t


of J. Andr~ the plane ~ is a vector space of dimension 2d over i t s kernel K

and a l i ne through the o r i g in is a subspace of dimension d. (Note that we are

viewing ~ as an a f f i ne plane.) T. G. Ostrom was the f i r s t to ca l l d the

dimension of 7. (See Kal laher [ i 0 ; p. 73-74].)

Before proving our genera l iza t ion (Lemma 2.2) we need the fo l lowing resu l t .

LEMMA 2.1: Let V be a f i n i t e dimensional vector space over the f i e l d K, l e t H

be a subgroup of GL(V,K), and l e t W be the subspace of V consist ing of a l l

points f i xed by H. I f L is a f i n i t e f i e l d extension of K such that V is also

a vector space over L and H is a subgroup of GL(V,L) then

dimK(V)/dimK(W ) = dimL(V)/dimL(W ).

PROOF. Since W consists of the points of V f i xed by H and since H e GL(V,L)

i t fo l lows that W is a subspace of V as a vector space over L. A standard

argument shows that dimK(V) = [dimL(V)][dimK(L) ] and dimK(W) =

[dimL(W)][dimK(L) ] . The lemma fo l lows.

LEMMA 2.2: Let ~ be a f i n i t e t r ans la t i on plane of dimension d over i t s kernel

G = GF(q), where q = pk with p an odd prime and k ~ 1. Let T be an elementary

2-group of Baer invo lu t ions of ~, and assume ITI = 2 t . I f L is a f i n i t e f i e l d

extension of K such that ~ is a vector space over L and T is a subgroup of

GL(~,L) then 2t+ l I d iv ides dimL(~ ).

PROOF: Ostrom [18] proves that 2 t ld . His proof ac tua l l y shows more; namely,

that

2 t + l = 2d/[dimK(W)] = [dimK(~)]/[dimK(W)] ,

where W is the subspace of ~ consist ing of a l l points in ~ f i xed by the group

T. By the preceding lemma

2 t+ l : [dimL(~)]/[dimL(W)] ,

and thus 2t+ll[dimL(~)],

3. SIMPLE GROUPS ACTING ON TRANSLATION PLANES

In th is section we consider the general s i t ua t i on of a f i n i t e noncycl ic simple

group act ing as a co l l i nea t i on group on a f i n i t e t rans la t i on plane of odd

cha rac te r i s t i c . We shal l show the smal lest possible dimension fo r the plane


is 4, the invo lu t ions of the group are Baer i nvo lu t i ons , and give some

informat ion about the representat ion of the gorup of the vector space.

The fo l lowing is s t ra ight forward.

LEMMA 3.1: Le t G be a f i n i t e noncycl ic simple group act ing as a co l l i nea t i on

group on a f i n i t e t rans la t i on plane ~ of dimension d over i t s kernel K = GF(q),

where q = pk with k ~ i and p a prime. The l i n e a r t rans la t i on complement

LTC(~) contains a subgroup G I with G 1 ~ G.

Thus we may assume, wi thout loss of genera l i t y , that under the hypothesis of

Lemma 2.1 the group G is contained in LTC(~). We shal l do so wi thout comment.

LEMMA 3.2: Under the hypothesis of Lemma 3.1, the group G cannot f i x a l i ne

pointwise.

PROOF: Assume G f i xes a l i ne ~ pointwise. The l i ne ~ cannot be the l i ne at

i n f i n i t y , since the group f i x i n g ~ pointwise is solvable. Since G f i xes the

o r ig in 0 the l i n e ~ must go through 0. The group G cannot contain e l a t i ons ;

fo r i f i t did then the group E of e la t ions in G with axis ~ must be normal in

G and thus E = G, cont rad ic t ing the fac t that G is nonsolvable. Thus G must

be a group of a f f i ne homologies with axis ~. But th i s is a cont rad ic t ion by

Coro l la ry 3.5 of LUneburg [12].

We now turn to some pos i t i ve information about the action of G on ~.

THEOREM 3.1: Let G be a f i n i t e noncycl ic simple group act ing as a co l l i nea -

t ion group on a f i n i t e t rans la t i on plane ~ of dimension d over i t s kernel

K = GF(q), where q = pk with p an odd prime and k ~ I . The fo l lowing s ta te-

ments hold:

( i ) Every invo lu t ion of G is a Baer invo lu t ion on ~.

( i i ) I f 2 a is the order of a maximal elementary abel ian 2-subgroup of G,

then 2 a ~ 4 and 2ald.

PROOF: Let T be an invo lu t ion in G. We f i r s t show that z commutes with at

leas t one of i t s n o n t r i v i a l conjugates. Assume not; that i s , assume that fo r

a l l ~ E G the elements �9 and - 1 do not commute. Then fo r a l l ~ E G the - i

commutator ~ TaT has odd order. (See Remark 2.1 of Fischer [3 ] . ) By

Glauberman's Z*-theorem (Dornhoff [ i ; Theorem 67.1]) t h i s impl ies T E Z(G),


a con t rad i c t i on to the s i m p l i c i t y of G. Thus there ex i s t s a n o n t r i v i a l con-

jugate ~ of T such tha t ~T = T~.

Assume z is not a Baer i n v o l u t i o n . Then z is a homology. I t cannot have axis

~ , since then <~> is a normal subgroup of G which con t rad ic ts the s i m p l i c i t y

of G. Thus T is an a f f i n e homology. Let ~ be the ( a f f i n e ) ax is of T and U

the center o f T. Then U E ~ and by Lemma 3.1 we may assume ~ = OV wi th

V E ~ . Hence ~ is also an a f f i n e homology, and Coro l l a ry 3.4 and Lemma 4.8

of LUneburg [12] imply ~ has axis @U and center V whi le ~T = T~ is an i nvo lu -

t o r y homology w i th ax is C and center O. But t h i s impl ies <z~> is a normal

subgroup in G, a con t rad i c t i on . Thus T is Baer. This proves statement ( i ) .

Since G is noncyc l i c simple the Brauer-Suzuki Theorem (Theorem 4.88 of

Gorenstein [6 ] ) impl ies every Sylow 2-subgroup has at l eas t two i n v o l u t i o n s .

Thus G contains elementary abel ian 2-groups of order 4. The remainder of

statement ( i i ) fo l lows from 0strom [18].

I f in the hypothesis of Theorem 3.1 the prime p is taken to be 2 then e i t h e r

statement ( i ) of the conclusion holds or the group G contains n o n t r i v i a l

a f f i n e e l a t i ons . I f the second p o s s i b i l i t y occurs then by the Hering-0strom

theorem (LUneburg [12; p. 178-179]), the group G = SL(2,2 s) f o r some s ~ 2 or

G = SZ(2 s) f o r a p o s i t i v e odd in teger s.

We turn now to looking at the representat ion of the simple group G on the

vector space ~ over the f i e l d K. We assume tha t G ~ s162162 tha t i s , tha t G

is a group of l i n e a r t ransformat ions on the vector space over K. Let T be an

i n v o l u t i o n in G. Then the i n v o l u t i o n T f i xes pointwise a Baer subplane of

which is a subspace of ~ having dimension d over the kernel K. (Recall t ha t

has dimension 2d as a vector space over K.) Since ~ has minimal polynomial 2

x - 1 i t fo l lows tha t T is represented by the 2d by 2d mat r ix o] - I d

where I d is the d by d i d e n t i t y ma t r ix , with r e spec t to a s u i t a b l e ba s i s .

Furthermore, i f ~ i s a vec tor space over a f i n i t e f i e l d ex tens ion k of K and

�9 E GL(V,L) then the argument of the previous paragraph s t i l l app l i e s s ince

the se t of f ixed po in t s of T i s the same.


We have proven the fo l low ing .

THEOREM 3.2: Let ~ be a f i n i t e t rans la t i on plane of dimension d over i t s

kernel K = GF(q), where q = pk with p an odd prime and k >_ 1. Let G be a

f i n i t e noncycl ic simple group contained in the l i n e a r t r ans la t i on complement

LTC(~), and l e t �9 be an invo lu t ion in G. The fo l lowing statements hold:

( i ) I f X is the character of the representat ion of G on ~ as a vector

space over K, then TX = O.

( i i ) I f ~ is a vector space over a f i n i t e f i e l d extension L of K with

G _< GL(~,L), and i f • is the character of the representat ion o f

G on ~ over L, then TXL = O.

4. THE SPORADIC SIMPLE GROUPS

We turn now to the s i t ua t i on in which a simple group acts i r r e d u c i b l y on the

underlying vector space of a f i n i t e t r ans la t i on plane. Mason [14] has consid-

ered th i s s i t ua t i on when the plane has even cha rac te r i s t i c and the group is of

untwisted Lie type with a s ingle root length defined over a f i e l d of charac-

t e r i s t i c two. We consider the s i t ua t i on when the plane has odd cha rac te r i s t i c

and the group, fo r the most par t , is a sporadic simple group.

Throughout th i s section we assume that ~ is a f i n i t e t r ans la t i on plane of

dimension d over i t s kernel K --GF(q), where q = pk with p odd and k >_ i . We

assume that G i s a simple group of co l l i nea t i ons in the l i n e a r t r ans la t i on

complement of ~ as a vector space over K. We w i l l make constant use of the

fo l lowing Theorem, which is e s s e n t i a l l y a restatement of Lemma 2.2 and Theorem

3.1.

THEOREM 4.1: Under the assumptions of the preceding paragraph there ex i s t s a

f i e l d extension L of degree s over K such that G is an abso lu te ly i r r educ ib le

subgroup of GL(2d' ,L) where d' = ds - I . Furthermore:

( i ) I f T is an invo lu t ion in G then over L i t s eigenvalues 1 and -1

occur with equal m u l t i p l i c i t y d ' .

( i i ) I f 2 t is the order of an elementary abel ian subgroup then 2 t

d iv ides d ' .


Although we have not been able to get much control over the size of s in gen-

e ra l , we know of no s i tuat ion in which i t cannot be taken to be 1 (see Mason

[14]). I t is the absolutely i r reducib le representation of G in GL(2d',L)

which we shall use throughout th is section. I f pIIGl and this representation

does not come from an ordinary one of that degree ( that is , i t l ies between

internal nodes of the Brauer tree at character is t ic p) we shall cal l i t pro-

per ly modular.

As an immediate consequence of these observations, we have the fol lowing.

THEOREM 4.2: Let ~ be a f i n i t e t ranslat ion plane of dimension d over i ts

kernel K = GF(q), where q = pk with p odd and k ~ 1, and l e t G be a simple

group of co l l ineat ions of ~ contained in the l inear t ranslat ion complement.

Assume G acts i r reduc ib ly on ~. I f the Sylow 2-subgroups of G are elementary

abelian, then p IGI and the representation of G must be properly modular.

PROOF: Let 2 t be the order of the 2-sylow subgroups of G. Then Theorem 4.1

implies 2t ld ' . I f the representation were ordinary then 2d' would divide iGi

by Corol lary 2 on p. 52 of Serre [19]. This is a contradict ion. The resu l t

fol lows.

Tlqe only groups which sat is fy the hypotheses of Theorem 4.2 are PSL2(q) with

3 < q and q z • (mod 8), SL2(2 n) with n ~ 2, the groups of Ree type, and J1

(iFeit [2]) . Of these groups the group J1 seems to be the only one for which

knowledge of the modular representations is complete (Janko [9], Fong [4]) .

For th is group, we have the fol lowing.

COROLLARY 4.2.1: Under the hypothesis of Theorem 4.2, i f G = J1 then p = 11,

the dimension d' of Theorem 4.1 equals 32, and the representation of J1 on

must be properly modular.

PROOF: The 2-Sylow subgroups of J1 are elementary abelian of order 8, so the

properly modular representation must have degree 2d' d i v i s ib le by 16. For p

any d iv isor of IGI other than 11 there are no such degrees, but at p = 11 the

ordinary representation of degree 120 sp l i t s into the d i rec t sum of two i r re -

ducible const i tuents: one of degree 56 coming from the ordinary representa-

t ion of that degree, and a new one of degree 64. This degree sa t is f ies con-

d i t ion ( i i ) of Theorem 4.1. Since involut ions have trace 0 at both degree 120 and 56, we cannot rule out degree 64 by looking at t h e t r a c e alone.


For the remainder of this section we apply Theorem 4.1 to the case where G is

one of the sporadic simple groups. The work wil l be divided into two subcases.

SUBCASE A: THE GROUP G IS A SPORADIC SIMPLE GROUP AND pIIGI

In this case, the representation of G comes from an ordinary irreducible one

of degree 2d'. The ordinary character associated with this representation

must take on the value O at each involution in G. The only Mathieu groups

satisfying this requirement are Mll with 2d' = 16 and M23 with 2' = 896

(see James [8]). As for the other sporadic groups, their character tables are

not all available in the literature. McKay [15] has the tables of the f i r s t

two Janko groups, and Frame [5] gives the character table for the Higman-Sims

group. Lyons [13] and O'Nan [17] published the tables for their groups before

the groups had actually been pinned down. Of all these groups only Ly and ON

have ordinary irreducible representations which satisfy the trace requirement

of Theorem 4.1(i). A complete l i s t of the possibil i t ies for G and 2d' is

shown below in Table 1. The authors are indebted to R. L. Griess for provid-

ing the necessary information on those groups whose character tables are not

readily accessible. The nomenclature is Gorenstein's [6].

Sporadic Simple Group Possible Values for 2d'

Mli 24

M23 27 �9 7 J3 27"3"5 , 27" 19 J4 221" 33"5"7

Mc 27"7, 27"7" 11

He 210 �9 3 �9 7

Ly 28. 7" 67, 28"37"7" 11, 28. 7" 11" 31" 37 ON 29 �9 34- 5, 29" 73

.2 218 �9 7

F 5 215" 53. 5, 29. 73

F 3 214" 11" 19 F 1 246 �9 112. 133. 23" 31" 47

F22 } F24 Insufficient information

F 2

TABLE 1: Possible Dimensions of Irreducible Representations on Finite Translation Planes for Each Sporadic Simple Group G when p~IGl


I f a sporadic simple group does not occur in Table 1 then there is no possible

dimension for i t . This gives the fol lowing resul t .

THEOREM 4.3: Let ~ be a f ini te translation plane of dimension d over its

kernel K = GF(q), where q = pk with p an odd prime and k ~ I, and let G be a

simple sporadic group contained in the linear translation complement of ~.

i f G acts irreducibly on ~ as a vector space over K and pIIGj, then G cannot

be one of the following groups: J1' J2' M12' M22' M24' HS, Sz, Ru, .3, . I ,

F23.

SUBCASE B: THE GROUP G IS ONE OF THE MATHIEU GROUPS AND pjjG 1

As already discussed, i f G is a sporadic group not listed in Table 1 then

plJGI and the representation must be properly modular. At this writing only

the Mathieu groups and J1 seem to have been worked out completely. Corol lary

4.2.1 considers the group Jl" We turn now to the Mathieu groups.

The modular representations of the Mathieu groups have been completely deter-

mined by James ~ ] . We shall see that the only representations possible are

those which come from the ordinary ones mentioned above. We handle each group

in turn, using the fact that each has an elementary abelian subgroup of order

8, and re fer r ing throughout to James's paper [8]. We note f i r s t , however,

tl~at for p = 3, none of the Mathieu groups has a modular i r reducib le degree

d i v i s i b le by 16, so that character is t ic is not possible.

For the group Mll there are two primes to consider: 5, 11. In each case the

only degree d iv i s ib le by 16 is 16 i t s e l f , and in each case the representation

comes from the ordinary one of the same degree.

I f G = M12 then only the primes 5 and 11 need to be considered. In each case

the only admissible degree is 16, but for both primes one of the classes of

involut ions has nonzero character. Thus MI2 cannot arise.

I f G is M22 or M24, then for each possible prime (5, 7, 11 i f G = M22; 5, 7,

11, 23 i f G = M24) no degree of an irreducible modular representation is

divisible by 16. Thus both M22 and M24 cannot occur.

Assume G = M23; then the possible primes are 5, 7, 11, or 23. For p = 5 the

only degree divisible by 16 is 896, and the presentation comes from an


ordinary one of that degree.

For p = 7, in addition to the two ordinary representations of degree 896,

there is only one other of admissible degree: 208. But at characteristic 7,

the ordinary representation of degree 253 is the direct sum of the ordinary

representation of degree 45 and this modular representation of degree 208.

Thus, lett ing • be the (complex) character of degree 253, and ~1 and ~2 be the

(Brauer) characters of degree 45 and 208, respectively, we have X = ml + m2"

Using the ordinary character table for M23, W e see that i f ~ E G is an involu-

tion, then m2(~) z 2 (mod 7). This makes i t impossible for T to have trace 0

in i ts 7-modular irreducible representation of degree 208.

For p = 11 or 23 none of the degrees of the irreducible modular representa-

tions is divisible by 16.

We summarize this discussion in the following result.

THEOREM 4.4: Let ~ be a f in i te translation plane of dimension d over i ts

kernel K = GF(q), where q = pk with p an odd prime and k ~ 1, and let G be a

Mathieu group contained in the linear translation complement of ~. I f G acts

irreducibly on ~ as a vector space over K, and d' is the integer of Theorem

4.1, then either

( i ) G = Mll, p > 3, d' = 8

or

( i i ) G = M23, p = 5, 7, or p > 23, d' = 448.

In both cases the representation comes from the ordinary one of that degree.

REFERENCES

1.

2.

3.

4.

L. Dornhoff, Group Representation Theory, Parts A and B, Marcel Dekker, New York, 1971

W. Feit, The current situation in the theory of f in i te simple groups, Actes, Congr~s Intern. Math. 1970, p. 55-93

B. Fischer, A characterization of the symmetric groups on 4 and 5 let ters, J. Algebra 3 (1966), 88-98

P. Fong, On decomposition numbers of J1 and R(q), Symposia Mathematica, Vol. XII I , p. 415-422; Academic Press, New York, 1974


5. J. S. Frame, Computation of characters of the Higman-Sims group and i t s automorphism group, J. Algebra 20 (1972), 320-349

6. D. Gorenstein, Fini te Simple Groups, Plenum Press, New York and London, 1982

7. C. Hering, On the structure of f i n i t e col l ineat ion groups of projective planes, Abh. Math. Sem. Univ. Hamburg 49 (1979), 155-182

8. G. D. James, The modular characters of the Mathieu groups, J. Algebra 27 (1973), 57-111

9. Z. Janko, A new f i n i t e simple group with abelian Sylow 2-subgroups and i t s characterizat ion, J. Algebra 3 (1966), 147-186

10. M. J. Kallaher, Aff ine Planes with Transit ive Col l ineat ion Groups, North Holland, 1982

11. R. A. Liebler, Combinatorial representation theory and t ranslat ion planes, Fini te Geometries, p. 307-332; Marcel Dekker, New York, 1983

!2. H. L~neburg, Translation Planes, Springer-Verlag, Berlin/Heidelberg/New York, 1980

13. R. Lyons, Evidence for a new simple group, J. Algebra 20 (1972), 540-569

14. G. Mason, Irreducible t ranslat ion planes and representations of Chevalley groups in character ist ic 2, Fini te Geometries, p. 333-346; Marcel Dekker, New York, 1983

15. J. McKay, The non-abelian simple groups G, IGI < 106--character tables, Comm. in Algebra 7 (13)(1979), 1407-1445

16. E. Mendelsohn, Every group is the col l ineat ion group of some projective plane, J. Geometry 2 (1972), 97-106

17. M. E. O'Nan, Some evidence for the existence of a new simple group, Proc. London Math. Soc. (3) 32 (1976), 421-429

18. T. G. Ostrom, Elementary abelian 2-groups in f i n i t e t ranslat ion planes, Arch. der Math. 36 (1981), 21-22

19. J.-P. Serre, Linear Representations of Fini te Groups, Springer-Verlag, Berlin/Heidelberg/New York

Department of Pure and Applied Mathematics Washington State University Pullman, WA 99q 6% U.S.A.

(Eingegangen am 5. September q985)

(Revidierte Form am 23. Dezember q985)

Simple groups acting on translation planes

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