PRONORMALITY, CONTRANORMALITY AND GENERALIZED NILPOTENCY IN INFINITE GROUPS · 2003. 10. 15. ·...

Publ. Mat. 47 (2003), 389–414

PRONORMALITY, CONTRANORMALITY ANDGENERALIZED NILPOTENCY IN INFINITE GROUPS

Leonid A. Kurdachenko and Igor Ya. Subbotin

AbstractThis article is dedicated to some criteria of generalized nilpo-tency involving pronormality and abnormality. Also new resultson groups, in which abnormality is a transitive relation, have beenobtained.

1. Introduction

There is a variety of very practical characterizations of nilpotency infinite groups. However, we are losing a majority of them at the transi-tion to infinite groups. In this case, these characterizations are formingthe bases for distinct definitions of new classes of infinite generalizednilpotent groups. On the other hand, for every concrete characteriza-tion of finite nilpotent groups it is very interesting to determine thelimits in which these characterizations are still valid; or more generallyto define the boundaries in which a certain characterization leads to acertain class of generalized nilpotent groups. One of the most suitableclasses on which many different characterizations of nilpotency have beenextended is the class of finitely generated soluble groups. The main rea-son for this is the famous Robinson’s Theorem (see, for example, [RD 2,Theorem 10.51]) proving that if every finite factor-group of a finitely gen-erated soluble group G is nilpotent, then G is nilpotent. Among other“good” classes we can mention the class of Chernikov groups and theclass of minimax groups. These classes are not far from finite groups be-cause they satisfy very strong finiteness conditions. Nevertheless, thereare some classes of infinite groups, which are far enough from finitegroups and satisfy some distinct criteria of generalized nilpotency suchthat locally nilpotency, hypercentrality, the normalizer condition, andso on. The class of (locally nilpotent)-by-finite groups and the class

2000 Mathematics Subject Classification. 20F19, 20F24, 20E34.Key words. FC-hypercentral groups, FC-nilpotent groups, nilpotency criteria, con-tranormal subgroups, pronormal subgroups, abnormal subgroups, transitive relations.

390 L. A. Kurdachenko, I. Ya. Subbotin

of FC-groups serve as such examples. Recall, that a group G is calledan FC-group, if the conjugate class xG = {xg | g ∈ G} is finite foreach element x ∈ G. The class of FC-groups is unique in some certainsense: many fundamental properties (particularly concerning with somecriteria of nilpotency) of finite groups have been naturally extended toFC-groups. For example, we can mention the following results.

If every subgroup of an FC-group G is subnormal, then G is nilpo-tent (see, for example [SH 2]).

If an FC-group G is a direct product of its Sylow p-subgroups forall prime p, then G is hypercentral.

If an FC-group G satisfies the normalizer condition, then G ishypercentral.

If every maximal subgroup of an FC-group G is normal, then G ishypercentral.

If G is an FC-group and [G, G] ≤ Fratt(G), then G is hypercentral.

Last three statements are consequences of the following well-knownfact.

If for an FC-group G the factor-group G/ζ(G) is residually nilpo-tent, then G is hypercentral.

Therefore it is natural to find such generalizations of the class ofFC-groups, on which these and other conditions of hypercentrality couldbe extended.

Let G be a group. Put FC(G) = {x ∈ G | xG is finite}. It is easy tosee that FC(G) is a characteristic subgroup of G. The subgroup FC(G)is called the FC-center of G. A group G is an FC-group if and only ifG = FC(G).

Starting from the FC-center we can construct the upper FC-centralseries of a group G:

〈1〉 = C0 ≤ C1 ≤ · · · ≤ Cα ≤ Cα+1 ≤ · · · ≤ Cγwhere C1 =FC(G), Cα+1/Cα =FC(G/Cα), α

Pronormality and Generalized Nilpotency 391

Consider extensions of the criteria listed above on FC-nilpotent orFC-hypercentral groups.

Let G be a group, every subgroup of which is subnormal. Then Gis locally nilpotent (see, for example, [LS, Theorem 2.5.1]). Thereforeif G is FC-hypercentral, then G is hypercentral. Note that there ex-ists an FC-nilpotent non-nilpotent group G, every subgroup of whichis subnormal [SH 1]. For some subclasses of FC-hypercentral groupsthe situation looks much better: if, for example, G is a periodic FC-hy-percentral group with all subgroup subnormal, then G is nilpotent (itfollows from the main result of the paper [MW]); if G is a torsion-freegroup with all subgroup subnormal, then G is nilpotent [SH 3].

Let G be a group satisfying the normalizer condition. Then G islocally nilpotent [PB]. Therefore if G is FC-hypercentral, then G ishypercentral. Groups with the normalizer condition can be characterizedwith the help of the concept of the ascending subgroup. Recall, that asubgroup H of a group G is called ascending if there is an ascendingseries

H = H0 ≤ H1 ≤ · · · ≤ Hα ≤ Hα+1 ≤ · · · ≤ Hγ = Gsuch that Hα is a normal subgroup of Hα+1 for all α < γ. It is well-known that a group G satisfies the normalizer condition if and only ifevery subgroup of G is ascending [BR, Theorem 4.13]. The dual to theidea of an ascending subgroup is the concept of a descending subgroup.Let H be a subgroup of a group G. We construct the normal closureseries in the following way.

Put H1 = HG. If for an ordinal α the subgroups Hβ have been alreadyconstructed for all β < α, then put Ha = ∩β


A finite group is nilpotent if and only if it has no proper contra-normal subgroups.

Further we will also consider criteria of (locally) nilpotency involvingpronormal and abnormal subgroups. These subgroups introduced byP. Hall and R. W. Carter respectively in the study of nilpotency in finitegroups. Pronormal subgroups play the main role in generalizations bothnormality and Sylow properties (see for example, the survey [BB] andpapers [F], [C], [P], [M], [R 2], [W], [KS 1], [KS 2], [KS 3], [SK],[deFKS], [SE]). Recall that a subgroup H of a group G is pronormalin G, if H and Hg are conjugate in 〈H, Hg〉 for each element g ∈ G.A subgroup H of a group G is abnormal in G, if g ∈ 〈H, Hg〉 for eachelement g ∈ G.

We will also obtain some generalizations of the following criteria ofnilpotency of finite groups:

A finite group is nilpotent if and only if all its pronormal subgroupsare normal.

Note that in the class of locally nilpotent groups all pronormal sub-groups are normal [KS 3]. It is easy to see that the condition: “everypronormal subgroup of a group G is normal” is equivalent to the con-dition “G has no proper abnormal subgroups”. It follows from the factthat a normalizer of a pronormal subgroup is an abnormal subgroup(see some elementary properties of pronormality and abnormality listedat the beginning of Section 3). So we come to the following criterion:

A finite group is nilpotent if and only if it has no proper abnormalsubgroups.

As many others these criteria could be extended on FC-groups (seeProposition 3.4).

The following theorem is one of the main our results.

Theorem B. Let G be a group, A a normal abelian subgroup of G suchthat A ≤ FC(G) and G/A is an FC-group.

(1) If every pronormal subgroup of G is normal, then G is hypercentral.(2) If G does not include proper contranormal subgroups, then G is

hypercentral.(3) If G does not include proper abnormal subgroups, then G is hyper-

central.

For periodic groups these results could be straightened significantly.


Theorem C. Let G be a periodic group, the FC-center of which in-cludes a G-invariant subgroup H such that G/H is a hypercentral group.



central.

Corollary C1. Let G be an FC-nilpotent periodic group.



central.

A. Ballester-Bolinches and T. Pedraza [B-BP] have considered radicalperiodic groups with Chernikov Sylow p-subgroups for all primes p, whichhave no proper contranormal subgroups. The class of these groups theydenoted by B. Note that some results (in particular, Theorem 6) of thispaper can be extended on the class of periodic locally soluble groupswith Chernikov Sylow p-subgroups for all prime p (see Corollaries C2,C3).

It is interesting to admit that contranormality possesses the propertyof transitivity. However, simple examples show that normality, pro-normality, and abnormality in general are not transitive (see, for exam-ple, [BB]). The class of groups in which normality is a transitive relation(T -groups) is well studied now. These groups and some their generaliza-tions has been investigated by many authors. The most valuable resultsabout T -groups one can find in [RD 1]. The class of soluble groups withtransitivity of pronormality is fully described in the recent paper [KUS].As it follows from Theorem D below, the class of soluble groups in whichabnormality is transitive (STA-groups) is much wider and contains bothclasses of soluble groups with transitive normality and transitive pronor-mality (ST - and STP -groups respectively). More specifically, based onTheorem D and using [RD 1, Theorem 2.3.1] and [KUS, Theorem 2]one can easily check the following strict embeddings

{STP -groups} ⊂ {ST -groups} ⊂ {STA-groups}.


Theorem D. Let a group G includes a Dedekind normal subgroup Hsuch that G/H has no proper abnormal subgroup. Then the abnormalityis a transitive relation in G.

Corollary D1. Let a group G includes an abelian normal subgroup Hsuch that G/H has no proper abnormal subgroups. Then the abnormalityis a transitive relation in G.

Corollary D2. Let a group G includes an abelian normal subgroup Hsuch that G/H is hypercentral. Then the abnormality is a transitiverelation in G.

Corollary D3. Every metabelian group is a group in which abnormalityis a transitive relation.

Corollary D4. In soluble groups in which normality is a transitive re-lation abnormality also is a transitive relation.

2. On direct decompositions of ZG-modules connectedto some normal subgroups

Let G be a hypercentral group, H a normal subgroup of G, A aZG-module. Put ζZH(A) = {a ∈ A | ax = afor each x ∈ H}. It iseasy to see that ζZH(A) is a ZG-submodule of A. ζZH(A) is called theZH-center of A. Starting from the ZH-center, we can construct theupper ZH-central series of A

〈0〉 ≤ C0 ≤ C1 ≤ · · · ≤ Cα ≤ Cα+1 ≤ · · · ≤ Cγby the following rule: C1 = ζZH(A), Cα+1/Cα = ζZH(A/Cα), α < γ,ζZH(A/Cγ) = 〈0〉. The last term Cγ of this series is called the up-per ZH-hypercenter of A.

Let B, C be some ZG-submodules of A, B ≤ C. The factor C/B iscalled H-central (respectively H-eccentric) if CH(C/B) = H (respectivelyCH(C/B) �= H).Lemma 2.1. Let G be a hypercentral group, H a non-identity normalsubgroup of G, A a ZG-module, CG(A) = 〈1〉, B a ZG-submodule sat-isfying the following conditions:

(1) B and A/B are simple ZG-modules.(2) B is H-central, A/B is H-eccentric.

Then there is a ZG-submodule C such that A = B ⊕ C.Proof: Since G is a hypercentral group, H ∩ ζ(G) �= 〈1〉. Let 1 �= z ∈H ∩ ζ(G). Consider the mapping φz : a → a(z − 1), a ∈ A. The map-ping φz is a ZG-endomorphism by the selection of z, in particular, Imφz


and Kerφz are ZG-submodules of A. Moreover, Kerφz ≥ B. If we as-sume that Ker φz �= B, then Kerφz = A, that is z ∈ CG(A) = 〈1〉.Contradiction. Hence Kerφz = B, that is A/ Ker φz ∼= Im φz = C is aZG-submodule of A such that CH(Im φz) �= H. Moreover, C is simpleZG-submodule, therefore C ∩ B = 〈0〉, because C �= B. Since A/B is asimple ZG-module, C + B = A, that is A = B ⊕ C.

Lemma 2.2. Let G be a hypercentral group, H a non-identity normalsubgroup of G, H a ZG-module, CG(A) = 〈1〉, B a ZG-submodule sat-isfying the following conditions:

(1) B and A/B are simple ZG-modules.(2) A/B is H-central, B is H-eccentric.

Then there is a ZG-submodule C such that A = B ⊕ C.

Proof: Since G is a hypercentral group, H ∩ ζ(G) �= 〈1〉. Let 1 �= z ∈H∩ζ(G). The mapping φz : a → a(z−1), a ∈ A, is a ZG-endomorphismof A. In particular, Imφz and Kerφz are ZG-submodules of A. SinceCH(A/B) = H, Im φz ≤ B. If we assume that Im φz = 〈0〉, thenz ∈ CG(A) = 〈1〉. Contradiction. Hence Imφz �= 〈0〉. Since B is asimple ZG-submodule, Im φz = B. The ZG-factor A(z − 1)/B(z − 1)is a ZG-homomorphic image of A/B. Since CH(A/B) = H, CH(A(z −1)/B(z − 1)) = H. Since B is a simple ZG-submodule, from A(z − 1) =Im φz = B and CH(B) �= H we obtain the equality A(z− 1) = B(z− 1).Hence for every a ∈ A we have a(z − 1) = b(z − 1) for some b ∈ B. Itfollows that (a− b)(z − 1) = 0, that is a− b ∈ CA(z) or A = B + CA(z).Since CA(z) = Ker φz, CA(z) is a ZG-submodule. Since B is a simpleZG-submodule, then either B ∩CA(z) = 〈0〉 or B ≤ CA(z). In the lastcase A = CA(z), that is z ∈ CG(A) = 〈1〉, which is impossible. It followsthat B ∩ CA(z) = 〈0〉 and A = B ⊕ C where C = CA(z).

Let G be a hypercentral group, H a normal non-identity subgroupof G, A a ZG-module. A ZG-submodule C is called H-hypereccentric,if C has an ascending series

〈0〉 ≤ C0 ≤ C1 ≤ · · · ≤ Cα ≤ Cα+1 ≤ · · · ≤ Cγ

of ZG-submodules of A satisfying the following conditions:Cα+1/Cα is a simple ZG-module.Cα+1/Cα is an H-eccentric factor for every α < γ.


Corollary 2.3. Let G be a hypercentral group, H a non-identity normalsubgroup of G, A a ZG-module, having a finite ZG-composition series,CG(A) = 〈1〉, C an upper ZH-hypercenter of A. Then:

(1) A = C⊕E where E is a maximal H-hypereccentric ZG-submoduleof A.

(2) E includes every H-hypereccentric ZG-submodule of A, in partic-ular, this decomposition is unique.

Proof: Assertion (1) follows from Lemmas 2.1 and 2.2. Let B be anH-hypereccentric ZG-submodule of A. If (B + E)/E is non-zero, itincludes a non-zero simple ZG-submodule D/E. Since (B + E)/E ∼=B/(B∩E), D/E is ZG-isomorphic with some simple ZG-factor of B. Itfollows that CH(D/E) = H. On the other hand, (B+E)/E ≤ A/E ∼= C,that is CH(D/E) = H. This contradiction proves that B ≤ E.

Proposition 2.4. Let G be a group, A a normal abelian periodic sub-group of G such that A ≤ FC(G) and Q = G/CG(A) is hypercentral.Let H be a non-identity normal subgroup of Q, C an upper ZH-hyper-center of A. Then:

(1) A = C⊕E where E is a maximal H-hypereccentric ZQ-submoduleof A.

(2) E includes every H-hypereccentric ZQ-submodule of A; in partic-ular, this decomposition is unique.

Proof: Since A ≤ FC(G) and A is periodic, A has a local family offinite G-invariant subgroups {Aλ | λ ∈ Λ}. Since Aλ is finite, theZQ-submodule Aλ has a finite composition ZQ-series. By Corollary 2.3Aλ = Cλ × Eλ where Cλ is the upper ZH-hypercenter of Aλ (that isCλ = C ∩Aλ), Eλ is a maximal H-hypereccentric ZQ-submodule of Aλ,λ ∈ Λ. If Aλ ≤ Aµ, then by the isomorphism Aλ/(Aλ ∩ Eµ) ∼=ZQAλEµ/Eµ ≤ Am/Em ∼=ZQ Cµ we obtain that Aλ ∩ Eµ is a maxi-mal H-hypereccentric ZQ-submodule of Aλ, that is Aλ ∩ Eµ = Eλ.Put E = ∪λ∈ΛEλ. Then E is a G-invariant subgroup of A such thatE∩Aλ = Eλ, λ ∈ Λ. By construction E is an H-hypereccentric ZQ-sub-module of A. Since Aλ = Cλ×Eλ = (C∩Aλ)×(E∩Aλ), A = C×E. Inparticular, E is a maximal H-hypereccentric ZQ-submodule of A. Theproof of assertion (2) is the same as the relevant part of the proof of theprevious corollary.


3. Proofs of main results. The non-periodic case

For the reader convenience we are listing the following elementaryproperties of abnormal and pronormal subgroups (see, for example, [BB],[DH]). We are going to use some of them in this paper.

Let G be a group, U a subgroup of G.I. The following statements hold.

(1) If A is pronormal in G and A ≤ U , then A is pronormal in U .(2) If H is a normal subgroup of G, and H ≤ A, then A/H is

pronormal in G/H if an only if A is pronormal in G.

(3) If H is a normal subgroup of G and A is pronormal in G, thenAH/H is pronormal in G/H.

(4) If H is a normal subgroup of G and A is pronormal in G, thenNG(AH) = NG(A)H.

(5) If A is pronormal in G and A is subnormal in G, then A isnormal in G.

(6) If A is pronormal in G, then NG(A) is abnormal in G.

(7) If A is pronormal in G, and B is normal in G, then AB ispronormal in G.

(8) If G is a soluble group, then A is pronormal in G if and onlyif it satisfies the Frattini property. In particular, if A is pro-normal in G and H is a normal subgroup including A, thenG = NG(A)H.

(9) If A is abnormal in G, then A is abnormal in any intermediatesubgroup of G.

II. The following statements are equivalent.

(1) H is abnormal in G.

(2) H is pronormal in G and NG(H) = H.

(3) H is pronormal in G and NG(K) = K for any intermediatesubgroup K for H.

III. Let A be a subgroup of a group G and H be a G-invariant subgroupof A; then A is abnormal in G if and only if A/H is abnormalin G/H.

The following Lemma 3.1 is well-known (see, for example, [McL]).


Lemma 3.1. Let G be an FC-hypercentral group. If G is finitely gen-erated, then G is nilpotent-by-finite.

The following statements are almost obvious.

Lemma 3.2. Let G be a group, every subgroup of which is descending.

(1) If H is a subgroup of G, then every subgroup of H is descending.(2) If L is a normal subgroup of G, then every subgroup of G/L is

descending.(3) If U/V is a section of G, then every subgroup of U/V is descending.

Lemma 3.3. Let G be a group, every subgroup of which is descending.Then every locally (soluble-by-finite) subgroup H of G is locally nilpotent.

Proof: Let K is an arbitrary finitely generated subgroup of H and let Sbe a normal subgroup of finite index in K. By Lemma 3.2 K/S is a finitegroup, every subgroup of which is descending. In other words, everysubgroup of K/S is subnormal. It follows that K/S is nilpotent. Henceevery finite factor-group of K is nilpotent, so that K is nilpotent [RD 2,Theorem 10.51].

Proof of Theorem A: By Lemma 3.1 the group G is locally (nilpotent-by-finite). Therefore we may use Lemma 3.3 and obtain that G is locallynilpotent. Since G is FC-hypercentral, G is hypercentral.

Proposition 3.4. Let G be an FC-group.



central.

Proof: The factor-group G/ζ(G) is residually finite and periodic (see,for example, [RD 2, Lemma 4.31]). Since a finite group, in which allpronormal subgroups are normal (respectively, a finite group, which doesnot include the proper contranormal subgroups) is nilpotent, G/ζ(G) isresidually nilpotent. Since this factor-group is also periodic, G/ζ(G) islocally nilpotent. But a locally nilpotent FC-group is hypercentral (see,for example, [RD 2, Theorem 4.38]), so G/ζ(G), and therefore G, ishypercentral.


Recall that a ZG-module A is called to be monolithic if the intersectionof all non-zero ZG-submodules of A (the ZG-monolith of A) is non-zero.

Lemma 3.5. Let a group G includes a normal abelian periodic sub-group A satisfying the following conditions:

(1) A ≤ FC(G).(2) A is a monolithic ZG-module with the ZG-monolith M .(3) G/A is a hypercentral group.(4) G �= CG(M).

Then A has an ascending series

〈1〉 = A0 ≤ A1 = M ≤ A2 ≤ · · · ≤ Aα ≤ Aα+1 ≤ · · · ≤ Aγ = A

of G-invariant subgroups, every factor Aα+1/Aα of which is G-chief andisomorphic to M as a ZG-module for each α < γ.

Proof: Put H = CG(M). Since M is a simple ZG-submodule, condi-tion (1) gives that M is finite, so that H is a normal subgroup of finiteindex. Since G/H is nilpotent, it is a p′-group (see, for example, [DH,Corollary B.9.4]). By Proposition 2.4 A = C × E where C is an upperZH-hypercenter of A and E is a maximal H-hypereccentric ZG-sub-module of A. By the selection of H we have M ≤ C. Since M is aZG-monolith of A, the equation M ∩ E = 〈1〉 implies that E = 〈1〉. Inother words, A is ZH-hypercentral.

Let S/M = SocG(A/M) be a subgroup, generated by all minimalG-invariant subgroups of A/M . Choose an arbitrary minimal G-invariantsubgroup U/M of A/M . Condition (1) yields that U is finite, that is V =CG(U) is a normal subgroup of finite index. If we assume that V = H,then by Maschke’s Theorem (see, for example, [DH, Theorem A.11.4])U = M × W for some G-invariant subgroup W . Since A is monolithic,this is impossible. This contradiction proves that V �= H. In particular,H/V is a non-identity normal subgroup of the nilpotent group G/V ,therefore H/V ∩ ζ(G/V ) �= 〈1〉. Let V �= zV ∈ ζ(G/V ) ∩ H/V . Thenthe mapping φ : u → [g, u], u ∈ U , is a ZG-endomorphism of U . Sincez ∈ H, M ≤ Ker φ, Im φ = [g, U ] ≤ M . Since Kerφ �= 〈1〉, Ker φ = M ,Im φ = M , so that U/Ker φ = U/M ∼=ZG M . Hence every minimalG-invariant subgroup of S/M is isomorphic with M as a ZG-module.Since A ≤ FC(G), A has an upper G-socular series. Using the samearguments and transfinite induction, we prove this lemma.


Lemma 3.6. Let a group G includes a normal abelian periodic sub-group A satisfying the following conditions:

(1) A ≤ FC(G).(2) A is a monolithic ZG-module with the ZG-monolith M .(3) G/A is a hypercentral FC-group.(4) G �= CG(M).

Then G includes a subgroup S such that G = A�S. Furthermore, everyother complement to A in G is conjugate with S.

Proof: Put H = CG(M). Since M is a simple ZG-submodule, condi-tion (1) implies that M is finite, so that H is a normal subgroup of finiteindex. Since G/H is nilpotent, it is a p′-group and the center ζ(G/H) iscyclic (see, for example, [DH, Corollary B.9.4]). Put 〈gH〉 = ζ(G/H).Then M(g − 1) = [g, M ] is a ZG-submodule of M , so that M = [g, M ].By Lemma 3.5 A has an ascending series

〈1〉 = A0 ≤ A1 = M ≤ A2 ≤ · · · ≤ Aα ≤ Aα+1 ≤ · · · ≤ Aγ = Aof G-invariant subgroups, every factor Aα+1/Aα of which is G-chiefand isomorphic to M as ZG-module for each α < γ. In particular,[gAα, Aα+1/Aα] = Aα+1/Aα for each α < γ. It follows that [g, A] = A.

Suppose that CA(g) �= 〈1〉. Let 1 �= a ∈ CA(g) and put K = 〈a〉G.By (1) K is finite. Let

〈1〉 = K0 ≤ K1 ≤ · · · ≤ Ks = Kbe a series of G-invariant subgroups, every factor of which is G-chief.Put U = Kj+1, V = Kj . Since G/A is an FC-group, C/A = CG/A(gA)has finite index in G/A, and (3) yields that C is subnormal in G. Let

C = C0 � C1 � · · · � Cm−1 � Cm = Gbe the respective subnormal series. The finiteness of U/V guarantees theexistence of a minimal Cm−1-invariant subgroup B/V of U/V . It followsthat there are elements y1, . . . , yt such that U/V = B/V ×(y−11 By1/V )×· · ·× (y−1t Byt/V ). In turn, the finiteness of B/V implies that it includesa minimal Cm−2-invariant subgroup D/V . Again there are elementsw1, . . . , wr such that B/V = D/V × (w−11 Dw1/V )×· · ·× (w−1r Dwr/V ),and hence

U/V = D/V × (w−11 Dw1/V ) × · · · × (w−1r Dwr/V )

× (y−11 By1/V ) × (y−11 w−11 By1w1/V ) × · · · × (y−1t w−1r Dwryt)/V ).


Repeating the same arguments after finitely many steps we come toa minimal C-invariant subgroup E/V and elements x1, . . . , xq such that

U/V = E/V × (x−11 Ex1/V ) × · · · × (x−1q Exq/V ).

Since gA ∈ ζ(C/A), [g, E]/V is a C-invariant subgroup of E/V . Theneither [g, E] = V or [g, E] = E. Suppose that [g, E] = V and consider[g, x−1j Exj]. We have xjgx

−1j = gz where z ∈ CG(U/V ), therefore

[g, x−1j Exj ] = x−1j [xjgx

−1j , E]xj

= x−1j [gz, E]xj

= x−1j (z−1[g, E]z[z, E])xj

= x−1j V xj = V.

Since this is valid for each j, 1 ≤ j ≤ q, we obtain [g, U ] = V , which isimpossible. This contradiction proves that [g, E] = E, and it follows that[g, x−1j Exj ] = x

−1j Exj for each j, 1 ≤ j ≤ q. Thus K has an ascending

series

〈1〉 = L0 ≤ L1 ≤ L2 ≤ · · · ≤ Ln = Kof C-invariant subgroups, every factor Lj+1/Lj of which is C-chief and〈g〉-eccentric. On the other hand, the series 〈1〉 ≤ £ ≤ 〈a〉 ≤ K has arefinement, the all term of which is a C-invariant subgroup and everyfactor of which is C-chief. By Schreier Refinement Theorem factors ofthis series are isomorphic (as ZC-modules) to factors of the series

〈1〉 = L0 ≤ L1 ≤ L2 ≤ · · · ≤ Ln = K.

But each factor of this last series is 〈g〉-eccentric and we obtain acontradiction. This contradiction proves that CA(g) = 〈1〉.

Let x be an arbitrary element of C. Then gx = ga1 for some ele-ment a1 ∈ A. By the equation A = [g, A] there is an element a2 ∈ Asuch that a1 = [g, a2]. Thus we have x−1gx = ga1 = g[g, a2] = a−12 ga2.It follows that xa−12 ∈ CC(g) = R, so that G = AR. We have alreadyproved that A ∩ R = A ∩ CC(g) = CA(g) = 〈1〉, that is G = A � R.Let Q be another complement to A in C. Then ζ(Q) has an ele-ment e such that gA = eA. It follows that e = ga3 for some ele-ment a3 ∈ A. The equation A = [g, A] implies the equation a3 = [g, a4]for some element a4 ∈ A, so that e = ga3 = g[g, a4] = a−14 ga4. ThusCC(e) = CC(a−14 ga4) = a

−14 CC(g)a4 = a

−14 Ra4.


From e = ga3 we obtain that CA(e) = CA(g) = 〈1〉. Since e ∈ζ(Q), Q ≤ CC(e), and from the equation C = AQ we obtain CC(e) =Q(A ∩ CC(e)) = Q, that is Q = a−14 Ra4. Now we can apply Lemma 3of [RD 3].

Proposition 3.7. Let G be a group, A a normal abelian periodic sub-group of G such that A ≤ FC(G) and G/A is an FC-group.



central.

Proof: By Proposition 3.4 G/A is hypercentral. Proposition 2.4 impliesthe decomposition A = C ⊕ E where C is the upper ZG-hypercenterof A (that is the intersection of the upper hypercenter of G with A),E is a maximal G-hypereccentric ZG-submodule of A. Suppose thatG is not hypercentral. It follows that E �= 〈1〉. Let M be a minimalG-invariant subgroup of E, then M is G-eccentric, that is CG(M) �= G.Choose a G-invariant subgroup D of A which is maximal with respectto the following properties: C ≤ D and D ∩ M = 〈1〉. Then A/D isa monolithic ZG-module with the ZG-monolith MD/D. Furthermore,CG(MD/D) �= G. By Lemma 3.6 G includes a subgroup S/D such thatG/D = A/D �S/D and every complement to A/D in G/D is conjugateto S/D. Let now g be an arbitrary element of G. Then Sg/D is a comple-ment to A/D in G/D. Put H/D = 〈S/D, Sg/D〉. The equation G/D =(A/D)(S/D) implies that H/D = B/D � S/D = B/D � Sg/D whereB/D = A/D∩H/D is an H-invariant (and hence S-invariant) subgroupof A/D. However every S-invariant subgroup of A/D is also G-invariant.In other words, B/D is a ZG-submodule of A/D. As A/D, B/D ismonolithic and its monolith is H-eccentric. Using Lemma 3.6 we obtainthat every complement to B/D is conjugate with S/D. In particular, thesubgroups S/D and Sg/D are conjugate in 〈S/D, Sg/D〉. This meansthat the subgroup S/D is pronormal in G/D.

Suppose that every pronormal subgroup is normal. Then S/D isnormal in G. It follows that G/D = A/D �= S/D, in particular themonolith of A/D is central. This contradiction proves that E = 〈1〉 andtherefore G is hypercentral.

Suppose that G does not include proper contranormal subgroupsand let U/D = NG/D(S/D), V/D = A/D ∩ U/D. If aD ∈ V/D,xD ∈ S/D, then [aD, xD] ∈ S/D ∩ A/D = 〈1〉. In other words,


NA/D(S/D) = CA/D(S/D). If we assume that V/D �= 〈1〉, then V/Dincludes MD/D, in particular, MD/D ≤ CG/D(S/D). But this meansthat MD/D (and hence M) is central in G, which contradicts its se-lection. This contradiction proves the equation S/D = NG/D(S/D).We have already proved that S/D is pronormal in G/D. Since a nor-malizer of a pronormal subgroup is abnormal (see, for example, [DH,Lemma I.6.21]), the subgroup S/D is also abnormal. Then S/D is con-tranormal, and we obtain a contradiction. This contradiction proves (2)and obviously (3).

Proof of Theorem B: Let T be the periodic part of A. Choose an ele-ment u ∈ A\T and put U = 〈u〉G. Then t(U) is finite and U is finitelygenerated. It follows that there is a number k ∈ N such that E = Uk istorsion-free. In other words, A includes a non-identity finitely generatedtorsion-free G-invariant subgroup. Let p be a prime; then Ep is a non-identity G-invariant subgroup. If A/Ep is not periodic, then by above itincludes a non-identity torsion-free G-invariant subgroup.

Let M/Ep be a maximal torsion-free G-invariant subgroup of A/Ep.Then A/M is periodic. By Proposition 3.7 G/M is hypercentral. Letr = r0(E), then |E/Ep| = pr. We have E ∩M = Ep, that is |EM/M | =pr. Since EM/M is a normal subgroup of the hypercentral group G/H,the hypercenter with the number r includes EM/M . It follows that[E, G, G, . . . , G︸︷︷︸

r

] = [E,r G] ≤ M . On the other hand, [E,r G] ≤ E since E

is a normal subgroup, that is [E,r G] ≤ E ∩ M = Ep. Since this is validfor each prime p, [E,r G] ≤ ∩p∈PEp (here P is the set of all primes).Since E is a free abelian subgroup, ∩p∈PEp = 〈1〉. Hence [E,r G] = 〈1〉,that is the hypercenter with the number r includes E. In other words,if A is not periodic, then A∩ ζ(G) �= 〈1〉. Using Proposition 3.7 with thehelp of transfinite induction we can obtain that G is hypercentral.

4. Proofs of main results. The periodic case

Lemma 4.1. Let G be a group, H a normal periodic subgroup of G suchthat H ≤ FC(G).

(1) If every pronormal subgroup of G is normal, then H is hypercentral.(2) If G does not include proper contranormal subgroups, then H is

hypercentral.(3) If G does not include proper abnormal subgroups, then H is hyper-

central.


Proof: Let K be a finitely generated subgroup of H, L = KG. SinceH ≤ FC(G), L is finite. Let S be a Sylow p-subgroup of L, p ∈ Π(L).For every element g ∈ G the subgroup Sg is also a Sylow p-subgroup of L,in particular, it is a Sylow p-subgroup of 〈S, Sg〉. Since 〈S, Sg〉 ≤ L, it isfinite. It follows that S and Sg are conjugate. Thus S is a pronormal sub-group. If every pronormal subgroup of G is normal, S also must be nor-mal in G. Since it is valid for every prime p, L is nilpotent, so that H islocally nilpotent. Since H ≤ FC(G), H is a hypercentral subgroup. Sup-pose that G does not include proper contranormal subgroups. Since S ispronormal, NG(S) is abnormal (see, for example, [DH, Lemma I.6.21]),and hence contranormal. It follows that G = NG(S), i.e. S is normalin G. As above it follows that H is hypercentral.

Lemma 4.2. Let a group G includes an abelian normal p-subgroup A(p is a prime number) satisfying the following conditions:

(1) A ≤ FC(G).(2) G/CG(A) is a periodic p′-group.

If B is a finite G-invariant subgroup of A such that A = B×C for somesubgroup C, then there is a G-invariant subgroup E such that A = B×E.

Proof: Let L be a local family of finite G-invariant subgroups of A in-cluding B. This family exists since A ≤ FC(G). If L ∈ L, thenL = B × (L ∩ C). Since L is finite and CG(L) ≥ CG(A), G/CG(L)is a finite p′-group. By Maschke’s Theorem (see, for example, [RD 4, 4,8.1.2]) there exists a G-invariant subgroup D such that L = B ×D. Let

C(L) = {D | D is a G-invariant subgroup of L such that L = B × D}.

If L, M ∈ L, L ≤ M , Q ∈ C(M), then M = B × Q and L =B × (L ∩ Q). Since L ∩ Q is G-invariant, L ∩ Q ∈ C(L). Define themapping πM,L : C(M) → C(L) by the rule: QπM,L = Q ∩ L, Q ∈ C(M).Clearly, if L, M, K ∈ L and L ≤ M ≤ K, then πK,M,πM,L = πk,L. Bydefinition πL,L is an identity mapping for each L ∈ L. In other words,{C(L) | L ∈ L} is a projection set. Then there exists a complete pro-jection set P (see, for example, [KU, Chapter 55]). This means, thatif R, Q ∈ P, R ∈ C(L), Q ∈ C(M) where L, M ∈ L and L ≤ M , thenQ ∩ L = R. Since L is a local family, P is a local family too. It followsthat E = ∪P is a G-invariant subgroup. Clearly E ∩ B = 〈1〉. If a isan arbitrary element of A, then a ∈ L for some L ∈ L. We have nowL = B × R where R ∈ P . It follows that A = B × E.


Lemma 4.3. Let a periodic group G includes an abelian normal p-sub-group A (p is a prime) satisfying the following conditions:(A) A ≤ FC(G).(B) G/A is a hypercentral p′-group.

Then the following assertions hold:(1) If every pronormal subgroup of G is normal, then H is hypercentral.(2) If G does not include proper contranormal subgroups, then H is


central.

Proof: Since A ≤ FC(G), A has an ascending G-chief series〈1〉 ≤ A0 ≤ A1 ≤ · · · ≤ Aα ≤ Aα+1 ≤ · · · ≤ Aγ = A

with finite factors. Put M = A1. Then M is a minimal G-invariantsubgroup of A. In particular, M is an elementary abelian p-subgroup,that is M ≤ B = Ω1(A). By Lemma 4.2 B includes a G-invariant sub-group E such that B = M ×E. Let R be a maximal subgroup with thefollowing properties: E ≤ R and R ∩ M = 〈1〉. Put U/R = Ω1(AR).If we assume that U/R �= MR/R, then using Lemma 4.2 we come tothe existence of a non-identity G-invariant subgroup W/R such thatU/R = MR/R × W/R. However this contradicts the election of R.This contradiction proves the equation U/R = MR/R, in particular,U/R is finite. It follows that A/R is a Chernikov subgroup (see, forexample, [FL, Theorem 25.1]). Let D/R be the divisible part of A/R.Since G/R is periodic, (G/R)/CG/R(D/R) is finite (see, for example,[RD 2, Lemma 3.28]). It follows that G/R is hypercentral-by-finite. Bygeneralized Schur-Zassenhaus Theorem (see, for example, [DM, Theo-rem 2.4.5]) G/R = A/R � S/R where S/R is the Sylow p′-subgroupof G/R. Let g ∈ G and consider the subgroup Y/R =

〈S/R, (S/R)gR

〉.

Since Y/R is hypercentral-by-finite, Sylow p′-subgroups of Y/R are con-jugate [SE, Lemma 1.1], in particular, S/R and (S/R)gR are conjugate.It follows that S/R is pronormal.

Suppose that every pronormal subgroup is normal, so S/R is nor-mal in G/R. Hence G/R = A/R × S/R and G/R is hypercentral. Inparticular, [G, M ] ≤ R. On the other hand, M is a normal subgroupof G, therefore [G, M ] ≤ M , that is [G, M ] ≤ M ∩ R = 〈1〉. This meansthat M ≤ ζ(G). Using the same arguments by transfinite induction wecan prove that Aα+1/Aα is a G-central factor for each ordinal α < γ.Consequently G is hypercentral.


Suppose that G does not include proper contranormal subgroups.Then NG/R(S/R) is abnormal (see, for example, [DH, Lemma I.6.21]),and hence contranormal. It follows that G/R = NG/R(S/R), i.e. S/R isnormal in G/R. As above it follows that G is hypercentral.

Corollary 4.4. Let a periodic group G includes an abelian normalp-subgroup A (p is a prime), satisfying the following conditions:

(A) A ≤ FC(G).(B) G/A is a hypercentral group.

Then the following assertion hold:



central.

Proof: Since G/A is hypercentral, G/A = P/A × S/A where P/A athe Sylow p-subgroup of G/A, S/A is a Sylow p′-subgroup of G/A. Inparticular, P is a normal Sylow p-subgroup of G. By Lemma 4.3 thesubgroup S is also hypercentral, so that S = A × R where R is a (nor-mal) Sylow p-subgroup of G. In particular, G = P × R. Since R ishypercentral, G is also hypercentral.

Corollary 4.5. Let a periodic group G includes an abelian normal sub-group A satisfying the following conditions:

(A) A ≤ FC(G).(B) G/A is a hypercentral group.

Then the following assertion hold:



central.

Proof: Since A ≤ FC(G), A has an ascending G-chief series

〈1〉 ≤ A0 ≤ A1 ≤ · · · ≤ Aα ≤ Aα+1 ≤ · · · ≤ Aγ = A


with finite factors. Consider an arbitrary G-chief factor Aα+1/Aα ofthis series. Let Π(Aα+1/Aα) = {p} and denote by S a Sylow p′-sub-group of A. Then Aα+1/Aα is G-isomorphic with some G-chief factorof A/S. By Corollary 4.4 G/S is hypercentral, and hence Aα+1/Aα isG-central.

Corollary 4.6. Let a periodic group G includes a soluble normal sub-group A, satisfying the following conditions:(A) A ≤ FC(G).(B) G/A is a hypercentral group.

Then the following assertion hold:(1) If every pronormal subgroup of G is normal, then H is hypercentral.(2) If G does not include proper contranormal subgroups, then H is


central.

Proof: Since A is soluble, it has a finite series of G-invariant subgroups

〈1〉 ≤ A0 ≤ A1 ≤ · · · ≤ Ak = Awith abelian factors. Using Corollary 4.5 and ordinary induction we canprove that G is hypercentral.

Corollary 4.7. Let a periodic group G includes a residual finite normalsubgroup A, satisfying the following conditions:(A) A ≤ FC(G).(B) G/A is a hypercentral group.

Then the following assertion hold:(1) If every pronormal subgroup of G is normal, then H is hypercentral.(2) If G does not include proper contranormal subgroups, then H is


central.

Proof: By Lemma 4.1 A is hypercentral. Let F be a finite G-invariantsubgroup of A. Since A is residually finite, A includes a normal sub-group H of finite index such that H ∩ F = 〈1〉. Put L = CoreG(H). ByRemak’s Theorem A/L ≤ Πg∈GA/Hg, in particular, A/L is soluble. ByCorollary 4.6 G/L is hypercentral. Since FL/L is a finite normal sub-group of G/L, there is a number m such that [G, G, . . . , G︸︷︷︸

m

, F ] ≤ L. On


the other hand, F is a normal subgroup of G, so that [G, G, . . . , G︸︷︷︸m

, F ] ≤

F . This implies [G, G, . . . , G︸︷︷︸m

, F ] = 〈1〉. In other words, the hyper-

center of G with number m includes F . Since F is an arbitrary finiteG-invariant subgroup of A, the inclusion A ≤ FC(G) implies that theupper hypercenter of G includes A. Since G/A is hypercentral, it followsthat G is also hypercentral.

Proof of Theorem C: Let Z = ζ(H), then Z is also G-invariant andH/Z is residually finite (see, for example, [RD 2, Lemma 4.31]). ByCorollary 4.7 G/Z is hypercentral. Now we can apply Corollary 4.5which implies that G is hypercentral.

Proof of Corollary C1: Let

〈1〉 = G0 ≤ G1 ≤ · · · ≤ Gm = Gbe the upper FC-central series of a group G. We will use inductionby m. If m = 1, the result follows from Proposition 3.4. Let we havealready proved that G/G1 is hypercentral. By Theorem C G is alsohypercentral.

Corollary C2. Let G be a periodic locally soluble group with ChernikovSylow p-subgroups for all prime p. The following statements hold.



central.

Proof: By Kargapolov’s Theorem (see, for example, [DM, Theo-rem 2.5.14]) G includes a normal abelian divisible subgroup D such thatG/D is a residually finite group with finite Sylow p-subgroup for allprime p. Hence every finite factor-group of G is nilpotent, G/D is resid-ually nilpotent. But a locally finite residually nilpotent group is locallynilpotent. Thus G/D is a direct product of its finite Sylow p-subgroups,in particular, G/D is an FC-group. Hence D is an abelian subgroupwith Chernikov Sylow p-subgroups for all prime p, D ≤ FC(G). Nowwe can apply Corollary C1.

Lemma 4.8. Let G be a Chernikov group, D the divisible part of G. Ifevery proper G-invariant subgroup of D is finite and G does not includeproper contranormal subgroups, then D ≤ ζ(G).


Proof: Choose a finite subgroup F such that G = FD. Put H = FG,then H is a proper normal subgroup of G. If we suppose that H isinfinite, then H ∩D is infinite. Since H ∩D is G-invariant, H ∩D = D,that is D ≤ H. But in this case H = G. This contradiction proves thatH is finite. In particular, [G, G] is finite, and hence D ≤ ζ(G).

Lemma 4.9. Let G be a Chernikov group. Then G is central-by-finiteand nilpotent in each of the following cases:

(1) G has no proper contranormal subgroups.(2) G is a Baer group.(3) G is a Fitting group.

Proof: Denote by D the divisible part of G. Let G satisfies (1). Thesubgroup D has a series of G-invariant subgroups

〈1〉 = D0 ≤ D1 ≤ · · · ≤ Dm = D

satisfying the following condition: every proper G-invariant subgroup ofDj+1/Dj , 0 ≤ j ≤ m − 1, is finite. Corollary C1 implies that G is ahypercentral group. By Lemma 4.8 every factor of this series is central.The finite factor-group G/D is nilpotent, hence G is nilpotent. But anilpotent Chernikov group is central-by-finite (see, for example, [RD 2,Lemma 3.13]).

Suppose now that G satisfies (2). Then G is locally nilpotent (see, forexample, [LS, Theorem 2.5.1]). We have G = DK where K is a finitesubgroup. In particular, K is nilpotent. Since G is a Baer group, K issubnormal. By Lemma 4 of [HK] G is nilpotent, and G is central-by-finite.

Finally, if G is a Fitting group, then it is a product of a finitely manynormal nilpotent subgroups and by Fitting Theorem (see, for example,[RD 2, Theorem 2.18]) G is nilpotent.

Corollary C2. Let G be a periodic locally soluble group with ChernikovSylow p-subgroups for all prime p. The following statements hold.



central.


Corollary C3. Let G be a periodic locally soluble group with ChernikovSylow p-subgroups for all prime p. The following statements are equiva-lent.

(1) G does not include proper contranormal subgroups.(2) G = Xp∈Π(G)Gp where Gp is a central-by-finite nilpotent Chernikov

p-subgroup, p ∈ Π(G).(3) G is a Fitting group.(4) G is a Baer group.

Proof: (1) ⇒ (2). By Corollary 4 G is hypercentral, so that G =Xp∈Π(G)Gp, where Gp is a Sylow p-subgroup of G, p ∈ Π(G). The sub-group Gp is isomorphic with some factor-group of G, therefore Gp has noproper contranormal subgroups. By Lemma 4.9 Gp is central-by-finite.

(2) ⇒ (3) and (2) ⇒ (4) are obvious.(4) ⇒ (2) and (3) ⇒ (2) follow from Lemma 4.9.(2) ⇒ (1). Let H be a proper subgroup of G, then H = Xp∈Π(H)Hp,

where Hp is a Sylow p-subgroup H, p ∈ Π(H). Then there is a number psuch that, Hp �= Gp. Since Gp is nilpotent, HpGp �= Gp. It follows thatHG �= G.

5. A remark on transitivity of abnormality

In the group S4 the subgroup A4 is abnormal. The subgroup B =〈(12)〉 is maximal and therefore abnormal in A4. However, it is almostobvious that B is not abnormal in S4. So, as we mentioned alreadyin the Introduction, in general, abnormality is not a transitive relation.Nevertheless, the following result, which could be considered as a usefuland general criterion of abnormality, shows that the class of groups withtransitive abnormality is significantly broad. For example, as it followsfrom Lemma 5.1 below all metaabelian groups posses this property.

Lemma 5.1. Let a group G includes a normal subgroup H such thatG/H has no proper abnormal subgroups. If for any abnormal subgroup Bthe intersection B ∩ H is normal in H, then abnormality is a transitiverelation in G.

Proof: If G has no proper abnormal subgroups the statement is trivial.Let B be an abnormal subgroup of G. Then HB ≥ B is an abnormalsubgroup in G. Since the image of an abnormal subgroup is abnormal,HB/H is abnormal in G/H. However, G/H has no proper abnormalsubgroups. Therefore G = HB. Let C be an abnormal subgroup in B.


By the lemma conditions H ∩B = D is normal in B. Since D is normalin H, D is also normal in G = HB. An isomorphism B/D = B/(H ∩B) ∼= HB/H, implies that B/D has no proper abnormal subgroups. Asabove since C is abnormal in B, B = DC. Therefore G includes thenormal subgroup D and the abnormal subgroup B = DC, such that Cis abnormal in B. Let g be an arbitrary element of G. Since B = DCis abnormal in G, g ∈ 〈DCg, DC〉 = D 〈Cg, C〉. Then g = dc whered ∈ D, c ∈ 〈Cg, C〉. Since C is abnormal in B = DC, d ∈

〈Cd, C

〉.

From c ∈ 〈Cg, C〉 it follows that d = gc−1 ∈〈Cgc

−1, C

〉≤ 〈Cg, C〉. It

means that g = dc ∈ 〈Cg, C〉 and C is abnormal in G, because in G anabnormality is transitive.

Theorem D and its Corollaries D1, D2, and D3 follow directlyfrom Lemma 5.1. We can easily obtain Corollary D4 admitting thatany ST -group is metabelian [RD 1].

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Leonid A. Kurdachenko:Mathematics DepartmentDnepropetrovsk UniversityVulytsya Naukova 1349050 DnepropetrovskUkraineE-mail address: [email protected] address: [email protected]

Igor Ya. Subbotin:Mathematics DepartmentNational University9920 S. La Cienega BlvdInglewood, CA 90301U.S.A.E-mail address: [email protected]

Primera versió rebuda el 10 de setembre de 2002,

darrera versió rebuda el 28 d’abril de 2003.

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