On inert subgroups of a group

On Inert Subgroups of a Group.

DEREKJ.S. ROBINSON

Dedicato al Professor Guido Zappa in occasione del suo novantesimo compleanno.

ABSTRACT- A subgroupHof a groupGis calledinertifjH :H\H^gjis finite for allg inG. If every subgroup ofGis inert, thenGis said to beinertial. After giving an account of the basic properties of inert subgroups, we study the structure of inertial soluble groups. A classification is obtained for the groups which are fi- nitely generated or have finite abelian total rank.

1. Introduction.

A subgroupHof a groupGis said to beinertifjH:H\H^gjis finite for allg2G. This is equivalent to saying thatHis commensurable with each of its conjugates. Obvious examples of inert subgroups are normal subgroups, finite subgroups and subgroups of finite index. Somewhat less obvious is the fact that permutable subgroups are inert (Lemma 3.4 below): here a subgroup His said to be permutable in a group G if HKˆKH for all subgroupsKofG. Recently inert subgroups have received attention in the literature, mainly in the context of locally finite groupsÐsee [1], [2], [3], [6].

A groupGwill be calledinertialif every subgroup ofGis inert (such groups are termed totally inert in [3]). Clearly the class of inertial groups contains all finite groups, Dedekind groups and Tarski monsters, and so it is a highly complex class. On the other hand, Belyaev, Kuzucuoglu and SecËkin [3] have shown that no infinite locally finite simple group can be inertial.

In the present work, after a discussion of the basic properties of inert subgroups, we investigate the structure of soluble inertial groups. While there are many groups of this type, we are able to give complete de-

(*) Indirizzo dell'A.: Department of Mathematics, University of Illinois at Urbana-Champaign - Urbana, IL 61801, U.S.A; email: robinson@math.uiuc.edu.

scriptions in the cases where the group is finitely generated or has finite abelian total rank.

2. Results.

We begin by noticing some uncomplicated types of inertial groups.

Suppose that G is a group with an abelian normal subgroup Aof finite index. In addition, assume that eachg2Ginduces a power automorphism in A; thusa^gˆaⁿ for alla2Awherenˆn(a)2Z. ThenG is inertial.

For, if HG, then H\A/G and H=H\A, being finite, is inert in G=H\A. HenceHis inert inGandGis inertial.

More generally, a groupGis said to be inertialof elementary type if there are normal subgroupsFandA, withFA, such thatjFjandjG:Aj are finite,A=Fis abelian and elements ofGinduce power automorphisms inA=F: clearlyGis an inertial group.

Our first two results show that for wide classes of soluble-by-finite groups, the only inertial groups are those of elementary type.

THEOREM A. Let G be a hyper-(abelian or finite) group such that t(G)ˆ1. Then G is inertial if and only if G is abelian or dihedral.

Heret(G) is the unique maximum normal torsion subgroup ofG. Also a group G is called dihedral if there is an abelian group A such that G'Dih (A), where

Dih (A)ˆt;Ajt²ˆ1;a^t ˆa ¹; a2A

; the dihedral group on A.

THEOREM B. Let G be a finitely generated hyper-(abelian-by-finite) group. Then G is inertial if and only if it has a torsion-free abelian normal subgroup of finite index in which elements of G induce power auto- morphisms.

Thus, for the classes of groups in Theorems A and B, the only inertial groups are those of elementary type.

Inertial groups of non-elementary type

It is not difficult to find examples of inertial soluble groups which are not of elementary type. A simple example is the grouph it ^j AwhereAis of typep¹,h it is infinite cyclic anda^tˆa^1‡p, (a2A).

Recall that a soluble group withfinite abelian total rank(FATR) is a group G with a series 1ˆG0/G1/ /GnˆG where each factor is abelian and the sum of thep-ranks

X

p

r_p G_i‡1 G_i

is finite foriˆ0;1;. . .;n 1: in the summation herepis a prime or 0. Such groups have also been calledS₁-groupsand in the interests of conciseness we will use this terminology. (For background on S₁-groups see [5], Chapter 5.) Our interest centres on the class

S1F

of S₁-by-finite groups: within this class there are, it turns out, inertial groups of three types which are not elementary. We will now explain how these non-elementary types may be constructed.

Construction

LetDbe a divisible abelianp-group with finite positive rank and letF be a torsion-free abelian group, also of finite positive rank. By the spec- trumof a groupG

sp (G);

we mean the set of primes p for which G has a PruÈferp¹-group as an image.

I Split extension type

LetQbe an extension of a finite group by eitherFor Dih (F), and letQ act onDas a group of power automorphisms. Assume that

p62sp (C_Q(D)) and define

GˆQ^j D:

ThenGis anS1F-group and in fact it is inertial, as is proved in Lemma 10.1 below.

II Central extension types

Let QˆF or Dih (F) and regardDas a trivialQ-module. Since Dis divisible, Ext (Q_ab;D)ˆ0 and the Universal Coefficients Theorem shows

that

H²(Q;D)'Hom (M(Q);D);

whereM(Q) is the Schur multiplier ofQ.

Let d2Hom (M(Q);D) have infinite order (or equivalently, infinite image). IfD0<DandF0Fwe denote bydD0anddF0the natural induced homomorphisms fromM(Q) toD=D₀andM(F₀) toDrespectively.

Assume thatdsatisfies the following conditions.

(a) IfQˆF, (so thatM(Q)ˆF^F), then (F0^f)^dD0 is finite for all f 2F, wheneverd_D0;F0 ˆ0.

(b) IfQˆDih (F), thenp62sp (F₀) wheneverd_F0 is not surjective.

The existence of such homomorphismsdis a question that will be ad- dressed in §11. Next choose a central extension with cohomology classd,

d: D^>!G!!Q:

ThenGwill be called a group ofcentral extension type IIa orIIb, according to whether (a) or (b) applies. ClearlyGis anS1-group and it is shown in Lemma 10.2 thatGis inertial.

PrimaryS₁F-groups

Let G be any S₁F-group. Then G has a maximum divisible abelian torsion subgroupDandt(G=D)ˆt(G)=Dis finite. It follows from Theorem A that ifGis inertial, thenG=Dis inertial of elementary type, so it may be assumed thatD6ˆ1. It is shown in Lemma 6.2 thatGis inertial if and only ifG=Dp⁰is inertial for eachp2p(D). Furthermore,D=Dp⁰is the maximum divisible abelian torsion subgroup ofG=D_p⁰ and this subgroup is ap-group with finite rank.

AnS₁F-group whose maximum divisible abelian torsion subgroup is a p-group will be termed p-primary. Thus in order to characterize the in- ertial S1F-groups of non-elementary type, it suffices to describe those which arep-primary. This is accomplished in the following theorem, which is the main result of the paper.

THEOREMC. Let G be a p-primaryS1F-group. Then G is inertial if and only if either G is of elementary type or else it is an extension of a finite group by a group of one of the types I, IIa, IIb.

The proof of Theorem C is accomplished in §§6-10. Basic results on inert subgroups are presented in §3, while initial structural information

about inertial groups appears in §4. The proofs of Theorems A and B are given in §5.

Notation.

t(G) : the maximum torsion normal subgroup ofG.

Fit(G) : the Fitting subgroup ofG.

M(G) : the Schur multiplier ofG.

Dih (A) : the dihedral group onA.

p(G) : the set of primes dividing orders of elements ofG.

sp (G) : thespectrumofG, i.e., the set of primespfor whichGhas ap¹- image.

r_p(A) : thep-rankofA.

3. Basic Results on Inert Subgroups.

We begin with an easy result that is used constantly in verifying that a subgroup is inert.

LEMMA3.1. Let H and K be subgroups of a group G such that HK andjK:Hjis finite. Then H is inert in G if and only if K is.

PROOF. Letg2G. Suppose first thatjK :K\K^gjis finite; then so is jH(K\K^g):K\K^gj ˆ jH:H\K^gj:

AlsojH\K^g:H\H^gj jK:Hj, so thatjH:H\H^gjis finite.

Conversely, assume thatjH:H\H^gjis finite. Then so isjK:H\H^gj and hencejK:K\K^gjis finite.

COROLLARY3.2 ([3], Lemma 3). A subgroup which is commensurable with an inert subgroup of a group is itself inert.

PROOF. LetH;KGwithHinert inGandKcommensurable withH.

ThenH\K is inert inGby Lemma 3.1 sincejH:H\Kjis finite. By the same resultKis inert inG.

LEMMA3.3. If H and K are inert subgroups of a group G, then H\K is inert in G.

PROOF. Let g2G. Then jH:H\H^gj is finite and thus so is jH\K:H\K\H^gj. AlsojK:K\K^gjis finite, whence

jH\K \ H^g:H\K\H^g\K^gj is finite. ThereforejH\K :(H\K)\(H\K)^gjis finite.

On the other hand,the intersection of infinitely many inert subgroups need not be inert. For example, in a polycyclic group every subgroup is the intersection of subgroups of finite index, but not every subgroup of a polycyclic group need be inert: this is shown by the free nilpotent group of class and rank 2.

We note that, in contrast to Lemma 3.3, the join of a pair of inert subgroups need not be inert.

EXAMPLE. LetGˆhx;y;a;biwherex²ˆ1,y^xˆy ¹,a^xˆa ¹,b^xˆb, a^yˆab,b^yˆband [a;b]ˆ1. ThenTˆhx;yi 'Dih (Z) andAˆha;biis free abelian of rank 2. FurthermoreGˆT^j A:Notice thatTis generated by the finite subgroupsh ix andh i. Howeverxy Tis not inert inG: for by an easy calculationT\T^aˆCT(a)ˆ1:

An interesting source of inert subgroups is the permutable subgroups of a group.

LEMMA3.4. Permutable subgroups are inert.

PROOF. LetHbe a permutable subgroup of a groupGand letg2G.

Then

H^{h ig} ˆH^{h ig} \(H gh i)ˆH(H^{h ig} \h i)g and

jH^{h ig} :Hj ˆ jH^{h ig} \h ig :H\h ij;g

which is finite unlessghas infinite order andH\h i ˆg 1. But under these circumstances H^gˆHby a result of Stonehewer [7]. Thus in any event jH^{h ig} :Hjis finite and

jH:H\H^gj ˆ jHH^g:H^gj jH^{h ig} :Hj;

sojH:H\H^gjis finite.

COROLLARY 3.5. The intersection of finitely many permutable sub- groups is inert.

4. Inertial Groups.

In this section we collect some useful properties of inertial groups, mainly in the soluble case. The first result has already been noted in [3].

LEMMA4.1. FC-groups are inertial.

PROOF. LetGbe an FC-group and letHG,g2G. ThenCˆC_G(g) has finite index in G, so jH:H\Cj is finite. Since H\CH\H^g, it follows thatjH:H\H^gjis finite andHis inert inG.

REMARKS. 1.Inertial groups need not be FC-groups.

An obvious example is Dih (Z), the infinite dihedral group.

2. Finite extensions of FC-groups need not be inertial.

In fact there are finitely generated abelian-by-finite groups which are not inertial. An example is the group

Gˆx;a;bj[a;b]ˆ1ˆx³;a^xˆb; b^xˆa ¹b ¹

;

as follows from Proposition 4.2 below. Another example is given in [3].

3. The direct product of two inertial groups need not be inertial.

For example, Dih (Z)Zis not inertial, again by Proposition 4.2. (For another example see [3]).

4. There are nilpotent p-groups of class 2 which are not inertial.

For any primep, letGˆhx₁;x₂;. . .isatisfy the relationsg₃(G)ˆ1 and x^p_i ˆ1;(iˆ1;2;. . .). Then G⁰ˆZ(G)ˆc_ijji<jˆ1;2;. . .

ˆC say, wherec_ijˆ[x_i;y_j]. EvidentlyGis a nilpotentp-group of class 2.

DefineHˆhx1;x3;x5;. . .i. Then a simple calculation shows that H\H^x2ˆH\Cˆc_ijji;jodd

: SincejH:H\Cjis infinite,His not inert inG.

The next result is basic in the study of soluble inertial groups.

PROPOSITION4.2. LetAbe a torsion-free abelian normal subgroup of a groupG. Assume that every cyclic subgroup ofGis inert. Then elements ofGinduce power automorphisms inA, i.e., ifg2G, thena^gˆa^efor all a2A, whereeˆ 1. HencejG:C_G(A)j 2.

PROOF. Let 16ˆa2Aandg2G. First of all we show that [a;gⁱ]ˆ1 for somei>0, and for this purpose we may assume that gAhas infinite order.

Sinceh ig is inert inG, we know thath i=g h i \g h ig ^ais finite and hence g

h i \h ig ^a6ˆ1. Let 16ˆx2h i \g h ig ^a; thenxˆgⁱˆ(g^j)^aˆg^j[g^j;a] where i;j6ˆ0. SincejgAjis infinite, it follows thatiˆjand [a;gⁱ]ˆ1, as claimed.

Nexth ia is inert, soh i \a h ia ^g6ˆ1 and there existm;n6ˆ0 such that a^mˆ(aⁿ)^g. Now by induction onj

a^mj ˆ a^nj_g^j

;

and by the previous paragraph we haveaˆa^gifor somei>0, from which it follows that a^miˆaⁿⁱ. Hencemˆ n since a has infinite order. Conse- quently… †a^{n g}ˆa^enwhereeˆ 1, which implies thata^g ˆa^e. Thusginduces a power automorphism inA(and the sameewill suffice for alla2A).

The situation is less clear for abelian normal subgroups of inertial groups which are torsion. However the following result is sometimes useful.

PROPOSITION4.3. LetAbe an abelian normal torsion subgroup of a group G. If every subgroup of A is inert inG, thena^{h ig} is finite for all a2A,g2G.

PROOF. Suppose for the moment thata2Ahas prime orderpand put HˆDa^g2iji2ZE

:

Then a^{h ig} ˆhH[H^gi, so we may suppose H to be infinite. Since jH:H\H^gjis finite,H\H^g 6ˆ1. Let 16ˆx2H\H^g; then we may as- sume that

xˆa^f(g2)ˆa^h(g2)g

where 06ˆf;g2Zp[t], the polynomial ring. Then a^{f(g2) h(g2)g}ˆ1 and f(t²) h(t²)t6ˆ0 inZp[t]. This implies that a^{h ig} is finitely generated and hence finite.

In the general case suppose thatahas orderm>1 and letpbe a prime dividing m. Then Bˆ a^mp_{h ig}

is finite. Passing to the group ha;gi=Band using induction onm, we conclude thata^{h ig}B=Bis finite, whencea^{h ig} is finite.

The final result in the section highlights the special role played by di- visible abelian subgroups of an inertial group.

LEMMA4.4. Let G be an inertial group. Then G has a unique maximum divisible abelian torsion subgroup D and elements of G induce power automorphisms in D.

PROOF. LetLbe anyp¹-subgroup ofG. Ifg2G, thenjL:L\L^gjis finite, which shows thatLˆL^gandL/G. IfMis anyq¹-subgroup, then [L;M]ˆ1 sinceMcentralizes every finite subgroup ofL. ConsequentlyG contains a unique maximum divisible abelian torsion subgroup,Dsay. Ifdis ap-element inD, thendis contained in ap¹-subgroup and henceh id /G. It follows that elements ofGinduce power automorphisms inD.

5. Proofs of Theorems A and B.

We have developed enough of the theory of inertial groups to be able to prove the first two main results.

PROOF OF THEOREMA. Let G be a hyper-(abelian or finite) inertial group for whicht(G)ˆ1. We show thatGis abelian or dihedral.

Let N be a nilpotent normal subgroup of Gand let M be a maximal abelian normal subgroup of N. Ifa2Mandx2N, then, sinceMis tor- sion-free,a^xˆa^e, whereeˆ 1 by Proposition 4.2. Sincehx;aiis nilpo- tent, 1ˆ[a;_nx]ˆa^{(e 1)n} for some n>0, and it follows that eˆ1 and [a;x]ˆ1. HoweverMˆCN(M), soMˆNandNis abelian. Consequently the Fitting subgroupFˆFit(G) is abelian, and of course it is also torsion- free. WritingCˆC_G(F), we conclude thatjG:Cj 2.

Assume thatF6ˆC. Then, sinceGis hyper-(abelian or finite), there is a non-trivial normal subgroup U=F of G=F such that UC and U=F is either finite or abelian. IfU=Fis finite,Uis centre-by-finite, soU⁰is finite and thus U is abelian. This implies thatUˆF, a contradiction. Hence U=Fis abelian, so thatUis nilpotent and againUˆF. ThereforeFˆC andjG:Fj 2.

If FˆG, then G is abelian. Otherwise F6ˆG and Gˆht;Fi; thus f^t ˆf ¹, (f 2F), andt²2F. Hence t⁴ˆ1 and thust²ˆ1 sinceF is tor- sion-free. FinallyGˆDih (F). The converse is clearly true.

COROLLARY5.1. A torsion-free hyper-(abelian or finite) group which is inertial is abelian.

PROOF OFTHEOREMB. LetGbe a finitely generated hyper-(abelian or finite) group which is inertial. It must be shown thatGpossesses a torsion-

free abelian normal subgroup of finite index in which elements ofGinduce power automorphisms. Notice that finitely generated groups with this structure are finitely presented. Thus, arguing by contradiction, we may assume the statement in the theorem is true for all proper quotients ofG, but false forGitself.

Ift(G)ˆ1, the result is a consequence of Theorem A, so we may as- sume t(G)6ˆ1. From this it follows thatGhas a non-trivial normal sub- groupAwhich is either finite or an abelian torsion group. Suppose thatAis finite. Then G=A satisfies the conclusion of the theorem and henceGis polycyclic-by-finite, which implies that it has a torsion-free normal sub- groupBof finite index. By Corollary 5.1 the subgroupBis abelian, and by Proposition 4.2 elements ofGinduce power automorphisms inB. By this contradictionAis an abelian torsion group.

Let 16ˆa2Aandg2G. Thena^{h ig} is finite by Proposition 4.3. Since G=Ais the product of finitely many cyclic groups,a^Gis finite. The argu- ment given above forAshows this to be impossible. The converse state- ment is clearly true.

6. InertialS₁F-Groups.

We begin the study of inertial S1F-groups with a discussion of the reduced groups. (For background onS1F-groups see [4], §5).

LetGb e anS₁F-group. Recall that the finite residualRofGis a di- visible nilpotent group. Its torsion-subgroupDis a divisible abelian with finite total rank, i.e., it is a direct product of finitely many PruÈfer p¹- groups. Clearly D is the unique maximum divisible abelian torsion sub- group ofG. IfDˆ1, thenGwill be calledtorsion-reduced.

The inertial S₁F-groups which are torsion-reduced are characterized in the following result.

PROPOSITION 6.1. Let G be a torsion-reduced S1F-group. Then the following are equivalent:

(i) G is inertial;

(ii) G is of elementary type;

(iii) there is a torsion-free abelian subgroup A of finite index in which elements of G induce power automorphisms;

(iv) G is finite-by-F or finite-by-Dih (F) where F is torsion-free abelian.

PROOF. It is either obvious or already known that (iii) )(ii))(i) and (iv))(ii). We show next that (i))(iii). LetGbe inertial. SinceGis tor- sion-reduced, its finite residualRis torsion-free. AlsoG=Ris reduced, so it has a normal torsion-free subgroup A=R with finite index in G=R. By Corollary 5.1 the subgroupAis abelian and by Proposition 4.2 elements ofG induce power automorphisms inA. Thus (iii) holds.

To complete the proof we show that (i) implies (iv). SinceGis inertial andt(G) is finite, we may suppose that the latter is trivial; the result now follows from Theorem A.

Primary groups

In the light of Proposition 6.1, we may confine our analysis to inertial S1F-groups which are not torsion-reduced. LetGb e anS1F-group and let Dbe its maximum divisible abelian torsion subgroup. ShouldDhappen to b e ap-group,Gwill be calledp-primary. We show next that it is sufficient to describe the inertialS₁F-groups which arep-primary.

LEMMA6.2. Let G be anS1F-group with maximum divisible abelian torsion subgroup D. Then G is inertial if and only if G=Dp⁰is inertial for all p inp(D). Moreover G=D_p⁰ is p-primary.

PROOF. Since the condition is certainly necessary, we assume that it holds forG. The groupG=Dp⁰ is inertial, so by Lemma 4.4 elements ofG induce power automorphisms inD=D_p⁰ and hence inD_p. Thus every sub- group ofDis normal inG.

Let HG: then H\D/G and in showing that H is inert in G we may suppose that H\Dˆ1. Let g2G and p2p(D). Then jHDp⁰ :HDp⁰ \ H^gDp⁰jis finite and thus so isjH:H\H^gDp⁰j. Sincep(D) is finite, it follows thatjH:H\Kjis finite, where

K ˆ \

p2p(D)

H^gD_p⁰ :

Ifk2K, thenkˆh^gpdpwherehp2Handdp2Dp⁰. Therefore, ifp6ˆq, we have

h^g_{q 1}

h^g_pˆdqd_p¹2H^g\Dˆ1;

which implies thatdpˆdqandhpˆhq. HenceKˆH^gand in consequence jH:H\H^gjis finite, so thatGis inertial.

7. Primary InertialS1F-Groups.

In this section we begin to analyze the structure of primary inertial S₁F-groups. LetGbe an inertialS₁F-group which is not of elementary type and denote by

D

its maximum divisible abelian torsion subgroup. SinceG=Dis of elemen- tary type by Proposition 6.1, we haveD6ˆ1. By Lemma 6.2 we can assume thatDis ap-group, i.e.,Gisp-primary.

There are three normal subgroups which play a prominent role in the analysis. By Proposition 6.1 there is a torsion-free abelian subgroupA=D with finite index in G=D in which elements of G induce power auto- morphisms. Also put

CˆC_A(D) and LˆC_G(A=D):

ThenA,CandLare normal inG,

DCAL and jG:Lj 2:

Since by Lemma 4.4 elements ofGinduce power automorphisms inD, there are two possible situations:

(i) [D;A]ˆ1, thecentral case;

(ii) [D;A]ˆA, thenon-central case.

There is a further dichotomy arising from the location of the sub- groupL:

(iii) LˆG, i.e.,A=DisG-central;

(iv) jG:Lj ˆ2.

It is easy to deduce from Proposition 6.1 that the groupG=Dis finite- by-torsion-free abelian in case (iii) and finite-by-dihedral (on some torsion- free abelian group) in case (iv). Cases (iii) and (iv) are referred to as the non-dihedral caseand thedihedral caserespectively. Thus in all there are four cases to be dealt with.

The above notation withD,C,A,Lwill be maintained throughout this and the following two sections.

A reduction

In establishing the necessity of the conditions in Theorem C one very useful observation is that it is always possible to factor out by a finite

normal subgroup without disturbing the standard subgroups D,C,A,L.

The basis for this reduction is the next lemma, which assures us that in factoring out by a finite normal subgroup we remain in the same central or non-central and dihedral or non-dihedral cases.

LEMMA 7.1. Let F be a finite normal subgroup of the inertial S1F- group G. Then:

(i) DF=F is the maximum divisible abelian torsion subgroup of G=F;

(ii) C_G(DF=F)ˆC_G(D).

(iii) AF=DF'^G A=D, so AF=DF is torsion-free abelian;

(iv) C_G(AF=DF)ˆC_G(A=D).

PROOF. (i) Suppose thatE=Fis a divisible abelian torsion subgroup of G=F. SinceGis anS₁F-group,Eis a CÏernikov group and thereforeED=D is finite. ThusEDFandDF=Fis the maximum divisible abelian torsion subgroup ofG=F.

(ii) Ifg2C_G(DF=F), then [D;g] is finite; however it is also divisible, so [D;g]ˆ1.

(iii) SinceA=Dis torsion-free,A\FDand hence AF=DF '^G A=A\(DF)ˆA=D:

(iv) This follows from (iii).

COROLLARY7.2. (i) The central case applies toG=F if and only if it applies toG.

(ii) The dihedral case applies to G=F if and only if it applies to G.

As a consequence of the corollary we can make a first reduction.

LEMMA 7.3. It may be assumed (by factoring out by a finite normal subgroup) that L=D is abelian.

PROOF. SinceL=Dis centre-by-finite,L⁰D=Dis finite. Now elements of Ginduce power automorphisms inD, so we have [D;G⁰]ˆ1 and therefore L⁰Dis centre-by-finite. Hence (L⁰D)⁰is finite and may be factored out. This makesL⁰Dan abelian CÏernikov group. Factoring out by a further finiteG- invariant subgroup ofL⁰D, we can make this subgroup divisible and hence L⁰D.

8. The Central Case.

In the notation established in §7, we are faced with the situation where [D;A]ˆ1;

and henceG=CG(D) is finite. There are two sub-cases that must be treated separately.

The non-dihedral case

In this case we haveLˆGand thus we can assume thatG=Dis abelian by Lemma 7.3. The case where in addition [D;G]ˆDis disposed of by the next result.

LEMMA8.1. Assume that G=D is abelian and[D;G]ˆD. Then, modulo a finite normal subgroup, G has the form X^j D where X is abelian, its elements induce power automorphisms in D and p62sp (CX(D)).

From this we deduce:

COROLLARY8.2. The groupGis of split extension type (I).

For the subgroupXis abelian and torsion-reduced, so its torsion-sub- group is finite.

PROOF OFLEMMA8.1. SinceDˆ[D;G] andG=Dis abelian, it follows from [4, Theorem H] or [5, 10.3.6] thatH²(G=D;D) is a torsion group. By [5, 10.1.15]Gsplits overDmodulo some finiteG-invariant subgroup ofD. This subgroup may be factored out without affecting the hypotheses onG, b y Corollary 7.2. Therefore we may assume that

GˆXD and X\Dˆ1 whereXis abelian. Put

YˆC_X(D);

it remains to show thatp2=sp (Y). Assume that this is false.

SinceDis a non-trivial divisible abelianp-group andp2sp (Y), there exists a homomorphismu:Y!Dwith infinite image. Put

EˆY^1‡u and notice thatYDˆED.

Since [D;G]6ˆ1, there exists aginGsuch that [D;g]6ˆ1; thend^gˆd^a, (d2D), for somep-adic integera6ˆ1. NowGis inertial, sojE:E\E^gjis finite. Let u2E\E^g; thus uˆe1ˆe^g₂ˆe2[e2;g] where e_i2E. Since [E;g]DandE\Dˆ1, it follows that uˆe₁ˆe₂2C_E(g), and conse- quently

E\E^gˆCE(g):

Thereforej[E;g]j ˆ jE:C_E(g)jis finite.

Next lety2Y and writeeˆy^1‡u 2E. SinceY Z(G), we have [e;g]ˆ(yy^u) ¹(yy^u)^gˆ(yy^u) ¹y(y^u)^aˆ(y^u)^{a 1}:

Since [E;g] is finite, it follows that (Y^u)^{a 1}is finite. Buta6ˆ1, so this gives the contradiction thatY^uis finite, which completes the proof of Lemma 8.1.

We are left with the case where [D;G]ˆ1. Let T=Dbe the torsion- subgroup of the abelian groupG=D. ThenT=Dis finite, soTis centre-by- finite and thusT⁰ is finite. Factoring out byT⁰, we can assume thatTis abelian. SinceTis a CÏernikov group, we can make it divisible by factoring out another finite normal subgroup. Hence TˆD. Therefore we can suppose thatG=D is a torsion-free abelian group.

At this point it is convenient to have the following technical result at our disposal.

LEMMA 8.3. Assume that [D;G]ˆ1 and G=D is abelian. Suppose further that DBG where B is abelian. Then [B;g]is finite for all g in G.

PROOF. SinceDis divisible,BˆDFwhereFis abelian. NowFis inert inG, sojF:F\F^gjis finite for allg2G. NextF\F^gˆCF(g) b y an argument in the proof of Lemma 8.1. Thereforej[F;g]j ˆ jF:C_F(g)jis fi- nite. Finally, [B;g]ˆ[F;g] since [D;G]ˆ1.

We can now complete the analysis of the central/non-dihedral case by analyzing the central extension

D^>!G!!F;

whereFˆG=D, a torsion-free abelian group. Since H²(F;D)'Hom (M(F);D)

andM(F)ˆF^F(the exterior square), the groupGis determined by an element

d2Hom (F^F;D):

Now Im(d)ˆG⁰\DˆG⁰and this must be infinite - for otherwiseGwould be of elementary type. We will show thatdsatisfies the conditions for the central extension type IIa.

LetF₀FandD₀<D, and suppose thatd_D0;F0ˆ0. This means that, if B=D₀is the preimage ofF₀under the natural mapG=D₀!!F, thenB=D₀is abelian. By Lemma 8.3 applied to the group G=D₀, we conclude that [B;g]D0=D0 is finite for all g2G, which translates into (F0^f)^dD0 being finite wheref ˆgD2F. ThereforeGis a group of central extension type IIa, which concludes our discussion of the central/non-dihedral case.

The dihedral case

We have now to deal with the situation where [D;A]ˆ1 and jG:Lj ˆ2;

i.e., the central/dihedral case. Thus elements inGnLinduce the inverting automorphism inA=D. An essential role in our analysis of this case is played by the following technical result.

LEMMA8.4. Assume that[D;A]ˆ1andjG:Lj ˆ2. Suppose that B is an abelian subgroup satisfying DBA. If p2sp (B=D), then each element of GnL induces the inverting automorphism in D, and in addition [D;L]ˆ1.

PROOF. SinceBis abelian, we haveBˆDFwhereFis torsion-free.

By hypothesisp2sp (B=D), so there is a homomorphismu:F!Dwith infinite image. WriteEˆF^1‡u, noting thatBˆDE:

Choose anyginGnL; then we claim that E\E^gˆe2Eje^gˆe ¹

:

To see this, suppose thatu2E\E^g; thenuˆe₁ˆe^g₂ˆe₂¹dwheree_i2E anddˆe^1‡g₂ 2D. SinceE\Dˆ1, we obtaine₁e₂ˆdˆ1, whenceuˆe₁ and e^g₁ˆe₁¹, as required. Since E is inert in G, it follows that jE^1‡gj ˆ jE:E\E^gjis finite. By the same argumentjF^1‡gjis finite.

Next letf 2F and puteˆf^1‡u2E. Therefore, writingdˆf^1‡g2D, we have

e^1‡gˆff^u(ff^u)^gˆff^uf ¹d(f^u)^aˆd(f^u)^1‡a;

where ginduces the automorphism x7!x^a in D, with aa p-adic integer.

SinceE^1‡gandF^1‡gare both finite, so is (F^u)^1‡a. HoweverF^uis infinite, so this can only mean thataˆ 1 andginduces inversion inD.

Finally, if`2L, theng`also induces inversion inD, which implies that [D; `]ˆ1 and [D;L]ˆ1. This completes the proof of Lemma 8.4.

We return to our analysis of the dihedral case. Let us assume for the present that in addition [D;L]6ˆ1, so that

Dˆ[D;L]:

Since L=D is abelian, it follows from [4, Theorem H] or [5, 10.3.6] that H²(L=D;D) is a torsion group, and becausejG:Ljis finite,H²(G=D;D) is also a torsion group. Therefore, modulo some finite subgroup of D, the groupGsplits overD, and by Corollary 7.2 we can assume that

GˆQ^j D

whereQis abelian-by-finite and inertial. ThusQis finite-by-Dih (A=D) b y Proposition 6.1.

NextAˆA\(QD)ˆ(A\Q)D, so Ais abelian since [D;A]ˆ1. Ap- plying Lemma 8.4 withBˆAand using the fact that [D;L]6ˆ1, we see that p62sp (A=D). Since A=D'A\QCQ(D) and jG:Aj is finite, it follows thatp62C_Q(D) and thereforeGis of split extension type I.

For the rest of the section we assume that [D;L]ˆ1:

Let T=Dbe the torsion-subgroup of the abelian group L=D. Then T is centre-by-finite, so, factoring out T⁰, we can assumeT to be abelian. Of courseTis CÏernikov, so factoring out one more time, we may assume that Tis divisible, i.e.,TˆD. ThusL=Dis torsion-free abelian. This allows us to replaceAbyL, which means that we now have the situation

[D;A]ˆ1; jG:Aj ˆ2 and

G=DˆDih (A=D):

There are two further sub-cases to be considered.

Case: A⁰ is finite

As usual we may assume thatAis abelian. WriteGˆht;Ai; thentinduces inversion inA=D. WriteAˆDF; thenjF:F\F^tjis finite, which by the usual argument shows thatF^t‡1is finite. NowF^t‡1/GsinceF^t‡1D, so we may factor out byF^t‡1and assume thattinduces inversion inF.

Next we have t²2A, which shows that t²t ²( modD) and t⁴2D.

Since A=Dis torsion-free,t²2Dand t² /G. Factor out by t² to get

t²ˆ1 andGˆh it ^j D. Nowd^tˆd^a, (d2D), for somep-adic integera. If p2sp (A=D), then Lemma 8.4 shows thataˆ 1 andG'Dih (A), which is of elementary type. Thereforep62sp (A=D).

We now have

Gˆt;Ajt²ˆ1; d^tˆd^a; f^tˆf ¹; f 2F

and GˆQ^j D where Qˆht;Fi ˆDih (F). Since p62sp (A=D)ˆsp (Q), the groupGis of split extension type I.

Case: A⁰ is infinite

In order to handle this case we need another lemma of a technical nature.

LEMMA8.5. Assume that[D;A]ˆ1,jG:Aj ˆ2and A⁰is infinite. Then DZ(G).

PROOF. Let g2Gand a₁;a₂2A. Then a^g_i ˆa^e_id_iwhere eˆ 1 and d_i2D. Hence

[a₁;a₂]^gˆ[a^e₁d₁;a^e₂d₂]ˆ[a^e₁;a^e₂]ˆ[a₁;a₂];

since [D;A]ˆ1 andA⁰D, which shows thatA⁰Z(G). SinceA⁰is infinite and elements of G induce power automorphisms in D, it follows that DZ(G).

By Lemma 8.5 we have acentral extension D^>!G!!QˆG=D;

say with cohomology classd2Hom (M(Q);D). AlsoQˆDih (A=D). Notice that Im(d)ˆG⁰\Dis infinite since it containsA⁰. We show thatdsatisfies the conditions of the central extension type IIb.

Let F0FˆA=Dand putD0ˆIm(dF0); assume thatD0<D. Then F0ˆB=D0is abelian. Ifp2sp(F0)ˆsp(B=D), then Lemma 8.4 shows that elements ofGnLinduce the inverting automorphism inD=D0. This is im- possible sinceDZ(G) by Lemma 8.5; hencep62sp (F₀), as required.

9. The Non-Central Case.

We now deal with the case where Dˆ[D;A]:

Also, of course,A=Dis torsion-free abelian andG=Ais finite.

Suppose first that we are in the dihedral case; thus jG:Lj ˆ2 and a^ga ¹( modD) fora2A,g2GnL. It follows thatA²[A;g]DG⁰D.

Since elements ofGinduce power automorphisms inD, we have [D;G⁰]ˆ1 and hence [D;A²D]ˆ1. Also,A²D=Dis torsion-free abelian andjG:A²Dj is finite. Consequently, if we replaceAbyA²D, we are again in the situation [D;A]ˆ1, i.e., we are in the central/dihedral case, which was dealt with in

§8: eitherGis of split extension type I or of central extension type IIb.

Now assume we have the non-central/non-dihedral case: thus LˆG andG=Dis abelian, whileDˆ[D;G]. It follows directly from Lemma 8.1 that, modulo a finite normal subgroup,Gis of split extension type I.

10. Sufficiency.

In order to complete the proof of the main result, Theorem C, we have to demonstrate that a group of type I, IIa or IIbis inertial. This is done in two lemmas.

LEMMA10.1. Let G be an extension of a finite group by a group of split extension type I. Then G is an inertialS₁F-group.

PROOF. We can assume thatGis of split extension type, so that GˆQ^j D:

hereDis a divisible abelianp-group of finite rank,Qis finite-by-For finite- by-Dih (F), with F a torsion-free abelian group of finite rank such that p62sp (C_Q(D)) andQacts as a group of power automorphisms onD. Clearly Gis anS1F-groupÐwhat needs to be shown is thatGis inertial.

Let HGand putD0ˆH\D/G. IfD0ˆD, thenHis inert since G=Dis inertial: assume that D₀<D. Then C_Q(D=D₀)ˆC_Q(D) and as a consequence we can assume that

H\Dˆ1:

By Proposition 6.1 there is a normal subgroupA of finite index such thatA=Dis torsion-free abelian with elements ofGinducing power auto- morphisms in it. SincejH:H\Ajis finite, it is enough to show thatH\A is inert inG. Thus it is permissible to assume thatHA, and henceHD=D is abelian.

Suppose that [D;H]6ˆ1. Then [D;H]ˆDand it follows from [5, 10.3.6]

and [5, 10.1.10] that, modulo a finite subgroup ofD, the complements ofDin

HDare conjugate toH. NowHDˆ(HD\Q)D, soHandHD\Qare con- jugate, which shows that we may assume thatHQ. HenceHis inert inQ.

Let x2Q and d2D. Then H\H^dˆC_H(d) and jH:H\H^dj j[H;d]j jd^Hj, which is finite. Thus jH:H\H^dj is finite. Also jH:H\H^xj is finite, as is jH^d:H^d\H^xdj, from which we deduce that jH\H^d:H\H^d\H^xdj is finite. Hence jH:H\H^d\H^xdj is finite, as must bejH:H\H^xdj. ThusHis inert in G.

Now assume that [D;H]ˆ1, so thatHDˆHD, which is abelian. If g2G, thena^ga^e( modD) fora2A, whereeˆ 1. Therefore

H\H^gˆfh2Hjh^gˆh^eg and

jH:H\H^gj jH^{g e}j;

sinceh7!h^{g e}is a homomorphism fromHtoD. Finally, sp (H)ˆsp (HD=D)sp (CQ(D));

sop62sp (H). ConsequentlyH^{g e}is finite and againHis inert.

LEMMA 10.2. Let G be an extension of a finite group by a group of central type IIa or IIb. Then G is an inertialS₁F-group.

PROOF. We may assumeGis of type IIa or IIb, so there is a central extension

D^>!G!!Q

withDa divisible abelianp-group of finite rank andQˆFor Dih (F), where Fis a torsion-free abelian group of finite rank. Let the cohomology class of the extension be

d2Hom (M(Q);D);

heredsatisfies the conditions for the type IIa or IIb.

LetHGand writeD0ˆH\D/G. It suffices to show thatH=D0is inert inG=D0. Suppose first that Im(dD0) is infinite; then we can pass at once to G=D0, i.e., we may assume thatH\Dˆ1. In addition we may suppose that HD=Dis abelian, so HDˆHDis abelian. If g2G, the usual argument shows thatjH:H\H^gj ˆ jH^{g e}j whereh^gh^e( modD), h2H, and eˆ 1. Also h7!h^{g e} is a homomorphism from Hto D. Let F0ˆHD=D. If Q is dihedral, then, since HD is abelian, dF0ˆ0 and p62sp (F0)ˆsp (H) and thusjH^{g e}jis finite.

If, on the other hand, Q is abelian, then eˆ1 and dF0ˆ0 where F0ˆHD=D. Therefore for anyg2Gthe group ((HD=D)^(gD))^dis finite, i.e., [H;g]ˆH^{g 1} is finite. ThusjH:H\H^gjis finite in both cases.

Now suppose that Im(d_D0) is finite, i.e., (G⁰\D)D₀=D₀is finite. Passing to the quotient group modulo (G⁰\D)D₀, we may assume thatD₀ˆ1ˆ

ˆ G⁰\D. Thusdˆ0 andGsplits over D. We now have GˆDQand H\Dˆ1. Also we may assume that QˆDih (F) and HDF. If t2QnF, then H\H^tˆh2Hjh^tˆh ¹

and jH:H\H^tj ˆ jH^t‡1j.

Nowp62sp (H) sincedˆ0, so the image of the homomorphism h7!h^t‡1, (h2H), is finite, i.e., jH^t‡1j is finite and again jH:H\H^tj is finite.

ThereforeHis inert inG.

This completes the proof of Lemma 10.2, and thereby Theorem C.

11. Existence Questions.

In this section we discuss the existence of the non-elementary inertial S1F-groups. There is little difficulty in constructing examples of groups of split extension type I for any given primep, in both the dihedral and non- dihedral cases. On the other hand, a group of central type IIa or IIbin- volves a torsion-free abelian group of finite rank which must satisfy some complex conditions. It is essential to produce explicit examples showing that groups of both types exist.

Constructing groups of central extension type

Letpbe an arbitrary prime andDa group of typep¹. Denote byQpthe additive group of p-adic rationals m

pⁿjm;n2Z

. The torsion-free abe- lian groupFthat we require is to be an extension ofZbyQpwith extension classeof infinite order,

e: C,!F!!Q_p whereCis infinite cyclic.

In fact such groups abound. For by standard homological calculations it can be shown that

Ext (Q_p;Z)'Z^_p=Z

whereZ^_pis the additive group ofp-adic integers. Sinceehas infinite order, the extension does not nearly split overC, i.e., there is no subgroup of finite index inFwhich splits overC.

A group of the above type may be constructed explicitly as follows. Let V be a 2-dimensional rational vector space with basisfu;vg. Letab e an irrational p-adic integer, written aˆa0‡a1p‡a2p²‡ ;(0a_i<p).

Define elementsv_iinV by

v0ˆvandv_iˆp ⁱ(v0‡(a0‡a1p‡ ‡a_{i 1}p^{i 1})u):

Thuspv_i‡1ˆv_i‡a_iu. Let

Fˆhu;v_ijiˆ0;1;2;. . .i:

ThenF=h i 'u Qpand it is not hard to show thatFdoes not nearly split over Cˆh i, sou Fis an example of the kind of group required.

We note some properties of abelian groups of this sort.

LEMMA11.1. Let C,!F!!Qpbe an abelian extension with extension class of infinite order, where C'Z. Then the following hold.

(i) M(F)ˆF^F'Q_p.

(ii) IfHF, thenHis either finitely generated or of finite index.

(iii) The groupFhas no subgroups isomorphic withQp.

PROOF. Let F=C'Q_p with C infinite cyclic. Then M(F) can be computed using the Lyndon-Hochschild-Serre spectral sequence for C,!F!!Q_p. Clearly E²₂₀ˆE²₀₂ˆE²₃₀ ˆ0; therefore M(F)'E²₁₁' 'QpZ'Qp.

Next letHF. Suppose first thatjF:HCjis finite. ThenH\C6ˆ1, otherwiseFwould nearly split overC. HencejHC :Hjis finite andjF:Hj is finite. IfjF:HCjis infinite, thenHC=Cis cyclic sinceF=C'Q_p. Hence His finitely generated.

Finally, (iii) follows directly from (ii).

The groupFwill now be used to construct examples of groups of types IIa and IIb. SinceM(F)'Q_p, there is a surjective homomorphism

d2Hom (M(F);D);

whereDis of typep¹. This determines a central extension d: D^>!G1!!F:

Next the inversion automorphism ofF acts trivially onM(F)ˆF^F, so it lifts to an automorphismtofG₁acting trivially onD. Now define

G2ˆh it ^j G1: Regarding the groupsG1;G2 we prove:

LEMMA11.2. The group G1is an inertialS1F-group of type IIa and G2

is an inertialS1F-group of type IIb.

PROOF. It is sufficient to check the validity of the conditions onFandd in the definitions of types IIa and IIb.

Suppose thatQˆG₂=DˆDih(F). IfF₀F andd_F0is not surjective, thendF0has finite order andF0cannot have finite index inF. HenceF0is finitely generated by Lemma 11.1, which shows that sp(F0) is empty. Thus G₂ is of type IIb.

Now assume thatQˆG₁=DˆF, and letD₀ <DandF₀F; thenD₀ is finite. Assume thatd_D0;F0 ˆ0. Thend_F0has finite order andjF:F₀jmust be infinite. HenceF0is finitely generated and therefore (F0^f)^dD0is finite for allf 2F. It follows thatG1 is of type IIa.

That the groupsG₁ andG₂are inertial follows from Lemma 10.2.

REFERENCES

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Sem. Mat. Univ. Padova,102(1999), pp. 151-156.

[4] J. C. LENNOX - D. J. S. ROBINSON, Soluble products of nilpotent groups, Rend. Sem. Mat. Univ. Padova,62(1980), pp. 261-280.

[5] J. C. LENNOX - D. J. S. ROBINSON, The Theory of Infinite Soluble Groups, Oxford 2004.

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Manoscritto pervenuto in redazione il 2 dicembre 2005.