Virtually free pro-p groups

W. Herfort∗ and P.A. Zalesskii†

February 18, 2010

Resume

Nous etudions la classe des groupes pro-p avec un sous-groupe ouverte etpro-p-libre engendre topologiquement par un ensemble finis. Nous montrons,que toute cette groupe admette une decomposition comme pro-p-produitamalgame sur un sousgroupe fini ou comme une extension pro-p-HNN avecgroupes associes finis. Par consequence, nous montrons, que toute cettegroupe est le groupe fondamental d’un graph finis des p-groupes finis.

Abstract

We prove that a finitely generated pro-p group G with an open freepro-p subgroup splits either as an amalgamated free pro-p product oras a pro-p HNN-extension over a finite p-group. Using this we showthat G is the pro-p completion of a fundamental group of a finite graphof finite p-groups.

Contents

1 Introduction 2

2 Preliminary results 42.1 Pro-p modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Pro-p modules and groups . . . . . . . . . . . . . . . . . . . . 72.3 Virtually free pro-p groups . . . . . . . . . . . . . . . . . . . . 7

∗Wolfgang Herfort would like to thank for the hospitality at the UNB during July 2006†Pavel Zalesskii expresses his thanks for support through CNPq

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3 HNN-extensions and HNN-embeddings 8

4 Lifting permutational representations to F×K 114.1 Induction engine . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Permutational extension criterion . . . . . . . . . . . . . . . . 14

5 Proof of the main theorems 17

6 Automorphisms 26

7 Strategy 307.1 Final proof in (old) section 4 . . . . . . . . . . . . . . . . . . . 40

8 Structure theorem – Example(s) 43

9 Lifting centralizers – Examples 43

10 1.2.10: Pavel’s suggestion for proving main result 47

11 Changes 51

12 What has been eliminated 5312.1 CF (t) = {1} for torsion elements t in a maximal subgroup of

F×K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1 Introduction

s-intro Let p be a prime number, and G a pro-p group containing an openfree pro-p subgroup F . If G is torsion free, then, according to the celebratedtheorem of Serre in [17] S 65 , G itself is free pro-p.

The main objective of the paper is to give a description of virtually freepro-p groups without the assumption of torsion freeness.

Theorem 1.1 t-main Let G be a finitely generated pro-p group with anopen free pro-p subgroup F . Then G is the fundamental pro-p group of afinite graph of finite p-groups of order bounded by |G : F |.

This theorem is the pro-p analogue of the description of finitely gener-ated virtually free discrete groups proved by Karrass, Pietrovski and Soli-tar [9] K-P-S 73 . In the characterization of discrete virtually free groups

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Stallings’ theory of ends played a crucial role. In fact the proof of the theoremof Karrass-Pietrovski and Solitar uses the celebrated theorem of Stallings in[18] St 70 , according to which every finitely generated virtually free groupsplits as an amalgamated free product / HNN-extension over a finite group.Note that a theory of ends has not been developed in the pro-p situation,although it has been initiated in [11] korenev . We prove a pro-p analogue

of Stallings’ Theorem and Theorem 1.1 t-main using purely combinatorialpro-p group methods combined with results on p-adic representations of finitep-groups.

Theorem 1.2 t-stallings Let G be a finitely generated virtually free pro-pgroup. Then G is either a non-trivial amalgamated free pro-p product withfinite amalgamating subgroup or it is a non-trivial HNN-extension with finiteassociated subgroups.

As a consequence of Theorem 1.1 t-main we obtain that a finitely gen-erated virtually free pro-p group is the pro-p completion of a virtually freediscrete group. However, the discrete result is not used (and cannot be used)in the proof.

V.A.Romankov proved in [15] Ro 93 that the automorphism group of a

finitely generated free pro-p group Aut(Fn) of rank n ≥ 2, is infinitely gener-ated. As a consequence one has that, despite the fact that the automorphismgroup Aut(Fn) of a free group of rank n embeds naturally in Aut(Fn), it isby no means densely embedded there! Nevertheless, Theorem 1.1 t-mainallows us to show that, surprisingly, the number of conjugacy classes of finitep-subgroups in Aut(Fn) is not greater than the corresponding number forAut(Fn).

Note that the assumption of finite generation in Theorem 1.1 t-main isessential: there is an example of a split extension H = F×D4 of a free pro-2group F of countable rank which can not be represented as the fundamentalpro-2 group of a profinite graph of finite 2-groups (see Example 5.10 t-4.3 ).

The line of proof is as follows. In Sect. 3 s-HNN we use a pro-p HNN-extension to embed a finitely generated virtually free pro-p group G in asplit extension E = F×K of a free pro-p group F and a finite p-group Kwith a unique conjugacy class of maximal finite subgroups. When K acts byconjugation freely upon F then our result is an immediate consequence ofProposition 14 in [6] paper3 . When this is not the case we then prove using

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an inductive argument the following theorem which connects the structureof any such group F×K with its representation on M := F/[F, F ].

Theorem 1.3 t-permlift Let E be a semidirect product E = F×K of afree pro-p group F of finite rank and a finite p-group K. Then the K-moduleM = F/[F, F ] is permutational if and only if F posesses a K-invariant basis.

This theorem gives an HNN-extension structure on E with finite basegroup. In particular, E and, therefore, G acts on a pro-p tree with finitevertex stabilizers. Using this and the machinery of the theory of pro-p groupsacting on trees we prove in Sect. 5 s-proofs Theorems 1.1 t-main and 1.2

t-stallings .

Basic material on profinite groups can be found in [19, 13] W 98, RZ1 2000 .Throughout the paper we make the following standard assumptions. Sub-groups are closed and homomorphisms are continuous. For x, y in a groupwe shall write yx := x−1yx and [x, y] := x−1y−1xy. For a subset A ⊆ G,

let 〈A〉abs denote the subgroup generated by A, 〈A〉 := 〈A〉abs the sub-group topologically generated by A, and (A)G the normal closure of A in G,i.e., the smallest closed normal subgroup of G containing A. For profinitegraphs we shall employ (standard) notations as found in [13] RZ1 2000 or

[14] RZ2 . The Frattini subgroup of G will be denoted by Φ(G). By Tor(G)we mean the subset of elements of finite order in G. For a finite p-group Glet socle(G) := 〈c ∈ Z(G) | cp = 1〉 denote the socle of G. Modules will befree Zp-modules of finite rank.

Acknowledgement

The authors are grateful to Karl Lorensen (Pennsylvania State University)for proof-reading our manuscript. We thank the referee for many helpfulsuggestions that led to considerable improvement of the paper.

2 Preliminary results

s-preliminaries

2.1 Pro-p modules

Theorem 2.1 t-Heller-Reiner (Heller-Reiner decomposition, (2.6) Theo-

rem in [3] heller-reiner ) Let G be a group of order p and M a Zp[G]-module,

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free as a Zp-module. Then

M = M1 ⊕Mp ⊕Mp−1,

where Mp is a free G-module, M1 is a trivial G-module and on Mp−1 theequality (1 + c+ · · ·+ cp−1)Mp−1 = {0} holds for any generator c of G.

Let G be a p-group. A permutation lattice for G (or G-permutationalmodule) is the direct sum of G-modules, each of the form Zp[G/H] for somesubgroup H of G. Note that a G-module M which is a free Zp-module isa permutation lattice if and only if G permutes the elements of a basis ofM . In particular, when H ≤ G and M is a G-permutation lattice it is anH-permutation lattice.

If G is of order p then Theorem 2.1 t-Heller-Reiner implies that M isa permutational lattice iff Mp−1 is missing in the decomposition for M iffM/(g − 1)M is torsion free for 1 6= g ∈ G.

Remark 2.2 rem-MG For a G-module M let φ : M → MG denote themap m 7→ (

∑g∈G g)m. By MG we denote the largest Zp[G]-epimorphic image

of M on which the induced action of G is trivial. We have MG = M/JM ,where J is the augmentation ideal. If G ∼= Cp and M is a free Zp[G]-modulethen MG = φ(M) = MG.

Lemma 2.3 l-gp-rg (Lemma 4 [6] paper3 ) For any finite group G, in-

tegral domain R and finitely generated R[G]-module M , the map φ is anepimorphism with kernel JM .

Corollary 2.4 c-sub With the assumptions of Theorem 2.1 t-Heller-Reinersuppose that M admits a Heller-Reiner decomposition M = M1 ⊕Mp. LetL be a free G-submodule of M and M/L be torsion free. Then, in a Heller-Reiner decomposition, Mp can be chosen to contain L as a direct summand.

Proof: Let φ be as in Remark 2.2 rem-MG . We argue by contradiction.Let M be a counter-example with minimal Zp-rank and M := M/L =Ap⊕Ap−1⊕A1 a Heller-Reiner decomposition. Since Ap is a free G-module,the preimage Ap of Ap in M decomposes in the form Ap = L⊕Up with Up afree G-module. We cannot have Ap 6= {0}, for otherwise Ap could serve as acandidate for Mp containing L, a contradiction.

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We show that M is a free G-module, i.e., M1 = {0}. Suppose firstthat there is an element x of a Zp-basis of M1 not contained in L. Denotepassing to the quotient modulo 〈x〉 by “tilde”. By the minimality assumptionM = L ⊕ K with K ∼= Ap−1 ⊕ A1. Denoting the preimage of K in M byN this means that M = L + N and, as L ∩ 〈x〉 = {0} we can conclude

from this that M = L ⊕ N . Now applying Theorem 2.1 t-Heller-Reinerto N one obtains a Heller-Reiner decomposition of M with a componentcontaining L, a contradiction. Hence M1 ≤ LG, so that in light of Remark2.2 rem-MG , by using Lemma 2.3 l-gp-rg we can deduce M1 ≤ φ(L).

Since M = Mp ⊕M1, one has φ(M) = φ(Mp) ⊕ pM1, implying that M1 ≤φ(L) ≤MG

p ⊕ pM1. As Mp ∩M1 = {0}, it follows that M1 = {0}.Since G is finite and L is a free G-module one has H1(G,L) = {0}.

Therefore there is a short-exact homology sequence

e-cohom 0→ LG →MG →MG → H1(G,L) = {0} (1)

associated to the short-exact sequence 0 → L → M → M → 0. Thenφ induces the Zp[G]-module epimorphism φ : M → MG given by φ(m) =∑p−1i=0 c

im for every m ∈M . Therefore MG can be considered as a submoduleof M and is torsion free since M is torsion free. As a consequence, in thedecomposition M = Ap−1 ⊕ A1, we must have φ(Ap−1) = {0} and so φ isthe monomorphism given by multiplication with p. For every x ∈ M/JMwe have φ(x) = 0 and since M/JM → M/JM is a Zp[G]-epimorphism wefind φ(M/JM) = {0}; hence M = 0. Thus M = Mp = L is a Heller-Reiner decomposition with a component containing L, a contradiction. TheCorollary is proved.

Lemma 2.5 l-cyclic Let C be a cyclic p-group, M a permutational C-

module and A the unique subgroup of order p in C. Then M/M1 is a freeC-module and M1 is a permutational C-module, where M1 is taken from theHeller-Reiner decomposition of the A-module M .

Proof: Since M is permutational M =⊕

H Zp[C/H], where H runs through(not necessarily distinct) subgroups of C. Then M1 =

⊕H 6={1} Zp[C/H] and

so M/M1 is a free C-module.

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2.2 Pro-p modules and groups

We shall need the following connection between a free decomposition of afree pro-p by Cp group G and a Heller-Reiner decomposition, Theorem 2.1

t-Heller-Reiner , of the G/F -module F/[F, F ], where F is a free pro-p opennormal subgroup of G of index p.

Lemma 2.6 l-modules-gen Let G be a split extension of a free pro-p groupF of finite rank by a group of order p. Then

(i) ([16] Sch 99 ) G has a free decomposition G = (∐i∈I(Ci ×Hi))qH,

with Ci ∼= Cp and all Hi and H free pro-p.

Here I is finite and each Ci is a representative of a conjugacy classof cyclic subgroups of order p in G. The subgroups Hi and H arecontained in F and CF (Ci) = Hi.

(ii) (Lemma 6 [6] paper3 ) Set M := F/[F, F ]. Fix i0 ∈ I and a gen-erator c of Ci0. Then conjugation by c induces an action of Ci0upon M . The 〈c〉-module M admits a Heller-Reiner decompositionM = M1 ⊕Mp ⊕Mp−1.

Moreover, the Zp-ranks of the three 〈c〉-modules satisfy rank(Mp) =p rank(H), rank(Mp−1) = (p−1)(|I|−1), and rank(M1) =

∑i∈I rank(Hi).

In particular, M is G/F -permutational if and only if |I| = 1.

2.3 Virtually free pro-p groups

We collect a few helpful results on pro-p groups.

Theorem 2.7 t-conjugacy (Theorem 4.3(a) [14] RZ2 ) Let G =∐ni=1Gi

be a free profinite (pro-p) product. Then Gi ∩ Ggj = 1 for either i 6= j or

g 6∈ Gj.

Lemma 2.8 l-fg-cfg (Corollary 3 [10] KZ 07 ) Every finitely generatedvirtually free pro-p group has, up to conjugation, only a finite number offinite subgroups.

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Lemma 2.9 l-conjugacy Let G = F×K be a semidirect product, with Ka finite p-group, F free pro-p of finite rank, and assume that every finitesubgroup of G is conjugate to a subgroup of K. Then for every subgroup Tof order p in K, every subgroup of finite order of CG(T ) is conjugate by anelement of CF (T ) to a subgroup of K.

Proof: Let R be a finite subgroup of CG(T ) and we can assume that itcontains T . By the hypothesis there exists f ∈ F such that Rf ≤ K. Then,T f ≤ K and for any generator t of T one has t−1tf ∈ K ∩ F , since F isnormal in G. As K ∩ F = {1} it follows that f ∈ CF (T ) as needed.

3 HNN-extensions and HNN-embeddings

s-HNN We introduce a notion of a pro-p HNN-extension as a generalization

of the construction described in [14] RZ2 , page 97.

Definition 3.1 d-HNN-ext Suppose that G is a pro-p group, and for afinite set I there are given monomorphisms φi : Ai → G for subgroups Aiof G. The HNN-extension G := HNN (G,Ai, φi, i ∈ I) is defined to be thequotient of G q F (I) modulo the equations φi(ai) = i−1aii for all i ∈ I.We call G an HNN-extension and term G the base group, I the set of stableletters, and the groups Ai and Bi := φi(Ai) associated.

One can see that every HNN-extension in the sense of the present defi-nition can be obtained by successively forming HNN-extensions, as definedin [14] RZ2 , each time defining the base group to be the just constructedgroup and then adding a pair of associated subgroups and a new stable letter.

There is an obvious universal property of G := HNN (G,Ai, φi, I). Givena pro-p group G, homomorphisms f : G → H, fi : Ai → H and mapsgi : Zi → H such that for all i ∈ I, all zi ∈ Zi and all ai ∈ Ai we have[gi(zi), fi(ai)] = 1 then there is a homomorphism ω : G → H which agreeswith f on G, with fi on Ai and with gi on Zi for every i ∈ I.

Theorem 3.3 t-HNN below is an HNN-embedding result – a refined pro-

p-version of the main theorem in [7] HZ 07 . We first prove it for semidirectproducts.

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Proposition 3.2 p-semidirect Let G = F×K be a semidirect product ofa free pro-p group F of finite rank and a finite p-group K. Then G can beembedded in a semidirect product G = E×K such that every finite subgroupof G is conjugate to a subgroup of K and E is free pro-p of finite rank.

Proof: We use induction on the number of conjugacy classes of finite sub-groups that are not conjugate to a subgroup in K. The base of induction istrivial. For the inductive step it suffices to show that G can be embeddedinto a semidirect product G of a finitely generated free pro-p group F and(the same) K with less conjugacy classes of finite subgroups that are notconjugate to a subgroup in K. So assume that L is a finite subgroup of Gnot conjugate to a subgroup of K. Let π : G −→ K be the canonical pro-jection and φ = π|L. Put G := HNN (G,L, φ) and observe that it is finitelygenerated.

For proving that G embeds in G we need to employ Theorem 1.3 in[1] Ch 94 , according to which G embeds in G if and only if the following setN of open normal subgroups intersects trivially: namely N is the set of allopen normal subgroups U of G such that there is a chain of normal subgroupsU = C0 < · · · < Cn = G with φ(L ∩ Ci) = φ(L) ∩ Ci and φ inducing theidentity on each (LCi ∩ Ci+1)/Ci for all i < n.

Let us show that every open normal subgroup U of G properly containedin F must belong to N . Consider the chain C0 := U , C1 := F and C2 := G.The conditions hold in the part below C1 = F since L∩F = φ(L)∩F = {1}.It is also trivial that φ(L ∩ C2) = φ(L) ∩ C2, since C2 = G. So we are leftwith showing that the homomorphism φ induced by φ on LF/F coincideswith the identity. If x ∈ LF/F with x ∈ L, then we have φ(x) = φ(x), andsince φ = π|L, φ(x) = π(x). By the definition of the projection π, if x = fk

with f ∈ F and k ∈ K, then π(x) = k. Hence φ(x) = π(x) = k = x, asdesired.

Note that π : G −→ K extends to G −→ K by the universal property ofan HNN-extension, so G is a semidirect product F×K of its kernel F withK. By Lemma 10 in [7] HZ 07 , every open torsion free subgroup of G mustbe free pro-p, so F is free pro-p. Let A be any finite subgroup of G. Then,by Theorem 4.2.(c) in [14] RZ2 , it is conjugate to a subgroup of the basegroup.

Having established the HNN-embedding result for semidirect products westate and prove it for arbitrary finitely generated virtually free pro-p groups.

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Theorem 3.3 t-HNN Let G be a finitely generated pro-p group possessingan open normal free pro-p subgroup F . Then G can be embedded in a semidi-rect product G = E×G/F such that every finite subgroup of G is conjugateto a subgroup of G/F and E is free pro-p. Moreover, G is finitely generated.

Proof: First set K := G/F and denote the canonical projection by π. FormG0 := GqK. By the universal property of the free pro-p product there is anepimorphism from G0 to K which agrees with π on G and the identity onK. As a consequence of the Kurosh subgroup theorem (see [13] RZ1 2000 ,Theorem 9.1.9), its kernel, say F0, is free pro-p and G0 = F0×K, where K isidentified with its image in G0. One observes that G0 is finitely generated,since G is. Now the result follows from Proposition 3.2 p-semidirect .

Definition 3.4 d-perm-ext Given a finite p-group K and a finite K-set X,

there is a natural extension of the action of K to the free pro-p group F (X).The semidirect product F (X)×K will be called the permutational extensionof F (X) by K.

Remark 3.5 rem-perm-ext Choosing representatives {Ai | i ∈ I} of theconjugacy classes of all point stabilizers and letting Zi ⊆ X be a set ofrepresentatives of orbits so that Kz = Ai for all z ∈ Zi, we can rewritethe K-set X in the form

⋃i∈I K/Ai × Zi with K acting on the cosets by left

multiplication and on the second factor trivially. Then G := F (X)×K hasa presentation F (

⋃i∈I Zi) q K modulo the relations [ai, zi] for all zi ∈ Zi

and ai ∈ Ai, with i running through the finite set I. The presentationshows that G is isomorphic to an HNN-extension in the sense of Definition3.1 d-HNN-ext , with all φi the identity on the respective group Ai, andwith the union

⋃i∈I Zi as the set of stable letters. We shall write G =

HNN (K,Ai, Zi, i ∈ I) – omitting the φi from the usual notation of the HNN-extension.

Let F := F (X). It is immediate that the induced representation ofG/F ∼= K on M := F/[F, F ] is K-permutational and can be written as adirect sum

M =⊕i∈I

Zp[K/Ai × Zi],

i.e., each K-submodule Zp[K/Ai] occurs with multiplicity |Zi|.

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One can present every permutational extension in the form G = HNN (K,Ai, Zi, i ∈I)q F (Z) with Ai 6= {1} for i ∈ I by letting I be representatives of all non-trivial associated groups and Z consists of those orbit representatives in Xwhose stabilizers are trivial.

4 Lifting permutational representations to F×Ks-permext

4.1 Induction engine

When G is the semidirect product G = F×K of a free pro-p group F anda finite p-group K let us denote for any subgroup H by NH the subgroupgenerated by all centralizers CF (x) of elements x of order p.

Lemma 4.1 l-ivstrengthen-new Let G be a semidirect product of a finitelygenerated free pro-p group F with a finite p-group K such that every torsionelement in G is conjugate to an element in K. Then the following statementshold true for every open normal subgroup H of G containing FΦ(K):

(i) F/NH is free pro-p and every torsion element in G/NH is conjugateto an element in KNH/NH .

(ii) CF/NH(tNH)/NH = CF (t)NH/NH for every torsion element t of G.

Proof: Observe also that it suffices to prove (ii) for all t ∈ K. Supposethat the lemma is false and that G is a counter-example with rank(F ) + |K|minimal.

Claim 1: K cannot be cyclic.

Suppose that K were cyclic. Then by Theorem 2.2 in [5] HZ 99 G =(∐t∈T CG(t)) q F0, where T is a suitable finite set of elements of order p

in G and F0 is free pro-p. By our assumption all elements of order p areconjugate to elements in K, so that Theorem 2.7 t-conjugacy shows that T

can contain a single element only. Therefore G = CG(t)qF0. Then G/NH =G/(CF (t))G = CK(t)q F0, showing that (i) and (ii) hold, a contradiction.

Claim 2: K 6∼= Cp × Cp. Morover, if P 6= F is a minimal normal subgroupof G containing FΦ(K) and y ∈ G, then G 6= 〈y, P 〉.

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For proving the first statement, recall that F has minimal possible rankamong all counter-examples with K ∼= Cp×Cp. Note that for every element

t ∈ K of order p in light of Lemma 2.6 l-modules-gen (i) there is a freedecomposition

e-CFT 〈t〉F = (〈t〉 × CF (t))q Ft (2)

with Ft free pro-p.We first show that we can restrict ourselves to assuming CF (K) = {1}.

Suppose that we have CF (K) 6= {1}. Then CF (K) is a free factor of CF (t),and its normal closure in F is normal in G. Therefore, using “tilde” to denotepassing to the quotient modulo (CF (K))F , we have in light of Eq.(2 e-CFT )

that ˜CF (t) = CF (t), so that NH = NH . By the minimality assumption onthe rank of F , CF (t)NH/NH = CF /NH

(tNH/NH). Hence

˜CF (t)NH/NH = CF /NH(tNH/NH).

Now the second isomorphism theorem applied to the normal subgroup (CF (K))Gimplies (i) and (ii), a contradiction.

Thus CF (K) = {1}.Next, set S = K ∩H. We show that for t ∈ K \ S the free factor Ft in

Eq.(2 e-CFT ) can be chosen such that CF (S) is a free factor of Ft. Since by

Lemma 2.9 l-conjugacy all subgroups of order p in CF (S)〈t〉 are conjugate,

we have CF (S)〈t〉 = 〈t〉 q F1 for some free pro-p subgroup F1 contained in

CF (S) (cf. Lemma 2.6 l-modules-gen (i)). By Lemma 2.6 l-modules-gen

(ii) M := F/[F, F ] = M1 ⊕Mp , where Mp is a free 〈t〉-module and M1 is atrivial 〈t〉-module. Moreover, the image L of CF (S) in M is a free 〈t〉-module

and so by Corollary 2.4 c-sub the submodule Mp can be chosen to containthe 〈t〉-submodule L as a direct summand, i.e. a free basis B of Mp can bechosen to contain a free basis of L. Then we can lift B to a subset X of Fsuch that X ∩ CF (S) is a basis of CF (S) and modify the choice Ft to be a afree pro-p group on X. Therefore Ft indeed can be chosen to contain CF (S)as a free factor.

It follows that F×〈t〉/(CF (S))G ∼= (CF (t) × 〈t〉) q Ft/(CF (S))Ft . Com-

paring with Eq.(2 e-CFT ) one can see that both (i) and (ii) hold, a con-tradiction. Thus K 6∼= Cp × Cp.

For proving the second statement of the claim, it is enough to observe thatP/Φ(K)F is either cyclic or the trivial subgroup of the elementary abelian p

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group K/Φ(K)F of order at least p3. So the images of P and y in K/Φ(K)Fcannot generate it. The second statement follows. The claim holds.

Claim 3: Let P be as in Claim 2, then (i) and (ii) hold for H = P .

During the proof of this claim we write “bar” for passing to the quotientmodulo NP , and we assume that a torsion element z in G/NP does not satisfyeither (i) or (ii). Since K is not cyclic, K ∩ P is a proper subgroup of K.Claim 2 implies the existence of a maximal subgroup Γ of G containing 〈P, z〉.

By the minimality assumption on rank(F ) + |K| and since |Γ∩K| < |K|,one deduces that z is a conjugate of an element in Γ ∩K. In the same veinwe can conclude that whenever z = x for some torsion element x ∈ G, thenCF (x) = CF (z). Hence such z cannot exist in Γ, a contradiction. Thus Claim3 holds.

We derive a final contradiction by showing that (i) and (ii) hold.

Since G is a counter-example, there exists an open normal subgroup Hof G containing FΦ(K) and y ∈ Tor(G/NH) for which either (i) or (ii) failsto hold.

Since K 6∼= Cp×Cp, H contains some subgroup P as described in Claim 2.During the rest of the proof we shall denote by “bar” passing to the quotientmodulo NP . By Claim 3, we have

e-NH NH = NH . (3)

Then, by the minimality assumption on rank(F )+|K| and since rank(F ) <rank(F ), the canonical image, say y, of y under the natural isomorphism

ψ : G/NH → G/NH (here we use Eq.(3 e-NH )) is a conjugate of an ele-ment in Tor(KNH/NH). Furthermore whenever there exists x ∈ Tor(G) withxNH/NH = yNH/NH , we have CF /NH

(y) = CF (x)NH/NH . Use of ψ−1 showsthat y is a conjugate of an element in K (we use the same letter to denotethe image of K in G/NH). If y = xNH/NH for some x ∈ Tor(G), then byClaim 3 (i) and (ii) hold for P so that we can conclude CF (x) = CF (x). Thelatter equality yields CF/NH

(y) = ψ−1(CF /NH(y)) = ψ−1(CF (x)NH/NH) =

ψ−1(CF (x)NH/NH) = CF (x)NH/NH , a contradiction.

Corollary 4.2 c-Ki cap H If for Ki ≤ K one has CF (Ki) 6= {1} then

Ki ∩H = {1}.

Proof: Suppose the statement is false. Then there is 1 6= x ∈ Ki ∩H withxp = 1. Then CF (Ki) ≤ CF (x) = CF (x) = {1}, a contradiction.

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4.2 Permutational extension criterion

Theorem 4.3 t-models Let G = F×K be a semidirect product, K a finitep-group, F free pro-p of finite rank, and assume that every torsion elementin G is conjugate to an element in K. Then G is a permutational extension.

Proof: Suppose that the theorem is false and that G is a counter-examplewith rank(F ) + |K| minimal. We claim that Z(G) = 1. If this were notthe case then in light of the minimality assumption we find that G/Z(G) =HNN (K, Ki, Zi, Z) where K := K/Z(G) (we certainly assuming that F isnot cyclic and so Z(G) ≤ K). Letting Ki be the preimages of the Ki we findthat G = HNN (K,Ki, Zi, Z), a contradiction. So Z(G) = {1}.Claim: There exists a subgroup U of index p in K and 1 6= t ∈ U withCF (t) 6= {1}

Assume that no such U exists. Then K acts by conjugation freely on F .Therefore Theorem 1 [6] paper3 implies that G is the free pro-p product ofa finite number of finite p-groups and a finitely generated free pro-p group.Since every torsion element is conjugate to an element in K we can concludethat G = K q F0 for some finitely generated free pro-p group F0. Then G isa permutational extension with I containig a single element i and Ti = {1},a contradiction. So the claim holds.

Fix for the rest of the proof a subgroup U of K having the propertystated in the Claim. Put H := UF . By the minimality assumption we havea permutational extension H = HNN (U,Ai, Zi, i ∈ I) q F (Z) for suitablefinite sets I and Z and Ai 6= {1} for all i ∈ I. Fix an element k ∈ K \ H.Conjugation with k permutes the finite subgroups of H, and, therefore, forevery Ai there is some ik such that Aik is conjugate to Aki in H. Thereforethere is an induced action of 〈k〉 upon I with kp acting trivially.

Let us observe that I decomposes as a disjoint union

I = I ∪ I1 ∪⋃p−1j=0 I

kj

p ,

I := {i ∈ I | CK(Zi) > Ai},I1 := {i ∈ I \ I | ∃h ∈ H,Aki = Ahi },

and Ip is a representative set of the set {i ∈ I |6 ∃h ∈ H,Aki = Ahi } modulothe action of K/U induced by conjugation. We stipulate that for all i ∈ Ipand for j = 0, . . . , p−1 the conjugate Ak

j

i is a representative of its conjugacyclass in H.

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By the minimality assumption the group G := G/NH = G/(⋃i∈I Zi)H

is a permutational extension, namely using the same letter K for its im-age in G we have G = HNN (K,Ki, Yi, i ∈ I) q F (Y0) (apply Lemma 4.1

l-ivstrengthen-new (i) to see that G meets the hypotheses of the theorem)

Let us remark that Ki∼= Cp for all i ∈ I as follows from Corollary 4.2

c-Ki cap H .

We want to match the decompositions of H and G and, to this end, wealter Z. When j ∈ I, then Lemma 4.1 l-ivstrengthen-new (ii) shows that we

have an epimorphism from CF (Kj) onto CF (Kj). Therefore we can definea retract ϕj : CF (Kj) −→ CF (Kj) and set Yj = ϕj(Yj). Finally, take aretract of F (Y0) inside F and select a basis Y in there with Y = Y0. Weobserve that F (Z) ∼= H/(U)H can be considered as a subgroup of indexp in HNN (K/U,KiU/U, Yi, i ∈ I) q F (Y0) containing Yi, Y0 and so by the

Kurosh subgroup theorem (see Theorem 9.1.9 in [13] RZ1 2000 ) can be

written as F (Z) ∼=∐j∈I F (Yj) q

∐p−1l=0 F (Y )k

l. So it allows to replace Z by⋃

j∈I Yj ∪⋃p−1l=0 Y

kl. Then

e-H-perm H = 〈U,Zi | i ∈ I〉 q∐j∈I

F (Yj) qp−1∐l=0

F (Y )kl

. (4)

For proving the Theorem it suffices to construct a permutational extensionG := HNN (K,Pj, Xj, j ∈ J) q F (X) and an isomorphism λ : G → G, thusobtaining a contradiction.

Let J := I ∪ I1 ∪ Ip ∪ I be a disjoint union. For j ∈ I, make Pj :=CK(CF (Aj)) and Xj := Zj; for j ∈ I1 ∪ Ip set Pj := Aj and Xj := Zj as well.If j ∈ I then set Pj := Kj and Xj := Yj. Finally let X := Y and with thesedata form G.

We define λ on⋃j∈J(Pj ∪ Xj) ∪ X ⊆ G to be the map that sends each

Pj and Xj in G identically to the same subgroup, respectively subset in G,and X onto Y . By means of the universal property of the permutationalextension G one can extend λ to become an epimorphism G −→ G.

Rewrite G as a permutational extension G = HNN (B,Ki, Xi, i ∈ I) qF (X), with base groupB := 〈K,Xi | i ∈ I∪I1∪Ip〉. Let B := HNN (B,Ki, Xi, i ∈I). Since H := λ−1(H) is open in G , one can apply the pro-p version of the

Kurosh subgroup theorem (see Theorem 9.1.9 in [13] RZ1 2000 ), taking into

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account that H contains the normal closure of F (X), to get a decomposition

H = (B ∩H)qp−1∐l=0

F (X)kl

.

Since H ∩Ki = {1} holds for each i ∈ I, Lemma 10 in [7] HZ 07 shows thatH ∩ B can be presented as a free pro-p product of conjugates of H ∩B and afree factor F0. Since all torsion elements in G are conjugate to elements in K,all torsion elements in H are conjugate to elements in U = K∩ H. Thereforeonly a single conjugate of H ∩ B can appear in this free decomposition ofH ∩ B, i.e., H ∩ B = (H ∩B)qF0. It is not difficult to see that (B)B ∩ H =(H ∩ B)H∩B, so F0 = B/(B)B = F (

⋃j∈I Xj); therefore we can choose F0 to

have a basis⋃j∈I Xj, so

H = 〈U,Xi | i ∈ I〉 q∐j∈I

F (Xj) qp−1∐l=0

F (X)kl

.

Comparison with Eq.(13 e-H-perm ) implies that the restriction of λ to H

is an isomorphism onto H. Since Z(G) = {1}, the epimorphism λ : G → Gis an isomorphism, a contradiction.

Theorem 4.4 t-permutational A semidirect product G = F×K of a finitelygenerated free pro-p group F with a finite p-group K is a permutational ex-tension of F by K (for a suitable subset of free generators for F ) if and onlyif the K-module M := F/[F, F ] is K-permutational.

Proof: Clearly, when G is a permutational extension then Remark 3.5rem-perm-ext implies that M is indeed K-permutational.

In light of Theorem 10.1 t-models for proving the theorem it suffices toshow that every torsion element x ∈ G is conjugate to an element in K.

Suppose there is G with rankF + [G : F ] minimal containing a torsionelement x not conjugate to an element in K. We first claim that such Gcannot be free pro-p-by-cyclic.

Suppose it is. By Theorem 2.2 in [5] HZ 99 G = (∐t∈T CG(t)) q F0,

where T is a suitable finite set of elements of order p in G and F0 is freepro-p. Let H be the unique open subgroup of G containing F as a subgroup

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of index p (recall that K ∼= G/F is cyclic). Then H = (∐t∈T ′ CG(t)) q F1,

where T ′ is a suitable finite set of elements of order p in H and F1 is freepro-p. Since M is H-permutational Lemma 2.6 l-modules-gen implies that

|T ′| = 1 so that H ∼= (Cp ×H1) q F1. Since K is cyclic we have T ⊆ H sothat we can conclude that |T | = 1, i.e. G = CG(t) q F0 and observe that

K ≤ GG(t). By Theorem 4.1 in [13] RZ1 2000 every torsion element of G isconjugate to a torsion element in one of the factors, hence we may assume thatx ∈ CG(t). The Zp-modules CF (t)/[CF (t), CF (t)] and CF (t)[F, F ]/[F, F ] =

M1 are isomorphic K-modules so that Lemma 2.5 l-cyclic implies that

CG(t) satisfies the assumptions of the theorem. Therefore the minimalityassumption implies that F0 = {1} and hence G = CG(t). We must now haveZ(G) ≤ K, else G ∼= Zp × K which is not a counter-example. Observingthis M turns out to be a permutational K/Z(G)-module and hence, by theminimality assumption there are g ∈ G and z ∈ Z(G) so that xg = kz, i.e.xg ∈ K, a contradiction. So G cannot be free pro-p-by-cyclic, as claimed.

For finishing the proof of the theorem we remark that there is a properopen subgroupH ofG containing 〈x〉F and, asM certainly isH/F -permutational,by the minimality assumption we can find h ∈ H with xh ∈ K, a contradic-tion.

5 Proof of the main theorems

s-proofsIn this section we shall use the notation and terminology of the theory

of pro-p groups acting on pro-p trees from [14] RZ2 . This also will be themain source of the references. In particular, when a pro-p group G acts on apro-p tree T we shall use the notation TG for the subset (in fact the subtree)of fixed points of T .

Lemma 5.1 l-actionfree Let G be a finitely generated pro-p group, Γ aconnected G-graph, and ∆ a connected subgraph of Γ such that ∆G = Γ.Then there exists a minimal set of generators X of G such that ∆ ∩∆x 6= ∅holds for all x ∈ X.

Proof: It is enough to prove the lemma under the additional assumptionthat G is elementary abelian. Indeed, using “bar” to denote passing to the

17

quotient modulo Φ(G), making Z a minimal generating set of G, suppose thatfor each z ∈ Z there exists a vertex vz ∈ ∆ with vz ∈ ∆∩ ∆z 6= ∅. Thereforethere exists fz ∈ Φ(G) with vzz ∈ ∆fz, so that the set X := {zf−1

z | z ∈ Z}is a minimal set of generators of G for which the assertion of the Lemmaholds.

Suppose that the lemma is false for an elementary abelian G. Then thereis a counter-example with d(G) minimal. Select a minimal generating subsetX of G. If d(G) = 1 then G is cyclic of order p, and, due to the connectednessof Γ, the set {∆g | g ∈ G} cannot be a partiton. Therefore there are x, y ∈ Gwith x 6= y and ∆x ∩ ∆y 6= ∅. Replacing X by {xy−1} shows that theconclusion of the lemma holds, a contradiction. Hence d(G) ≥ 2. Select anelement z ∈ X and let “bar” denote passing to the quotient modulo 〈z〉.Since d(G) = d(G) − 1, by the minimality assumption there is a subset Yof G which is a minimal set of generators of G and ∆ ∩ ∆y 6= ∅ is truefor all y ∈ Y . Let Y denote a section of Y in G. Then there are elementszy ∈ 〈z〉 such that ∆ ∩ ∆yzy 6= ∅ for all y ∈ Y . Set Y ′ = {yzy | y ∈ Y }.Since Γ can be viewed as a 〈z〉-space, ∆Y ′ ∩ ∆Y ′z 6= ∅ can be assumed bythe minimality assumption. This means that there exist y, y′ ∈ Y ′ such that∆y ∩ ∆y′z 6= ∅, so X = Y ′ ∪ {y−1y′z} would satisfy the assertion of thelemma, a contradiction.

Lemma 5.2 l-action Let a finitely generated pro-p group G act on a pro-ptree T , which contains a subtree D such that DG = T . Then there exists aminimal set of generators X of a retract H of G/〈Gv | v ∈ V (T )〉 in G suchthat D ∩Dx 6= ∅ for all x in X.

Proof: Let “bar” denote passing to the quotient modulo 〈Gv | v ∈ V (T )〉.By 3.5 in [14] RZ2 the quotient graph T := T/〈Gv | v ∈ V (T )〉 is a pro-p

tree. Applying Lemma 5.1 l-actionfree to G acting on T yields a subset Zof G such that Z is a minimal set of generators of G and for each z thereexists a vertex vz ∈ D such that vz z ∈ D∩ Dz 6= ∅ holds. Hence there existskz ∈ 〈Gv | v ∈ V (T )〉 with vzzkz ∈ Dkz ∩Dz and so vzz ∈ D ∩Dzk−1

z . Nowset X := {zk−1

z | z ∈ Z} and H := 〈X〉. Finally observe that by Corollary

3.6 in [14] RZ2 G is free pro-p, so that H is indeed a retract.

Lemma 5.3 l-freeedgeaction A pro-p group G acting on a pro-p tree Twith trivial edge stabilizers such that there exists a continuous section σ :

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V (T )/G −→ V (T ) is isomorphic to a a free pro-p product(∐

v∈V (T )/GGσ(v)

)q

(G/〈Gw | w ∈ V (T )〉).

Proof: This follows from the proof of Theorem 3.6 in [23] Z 95 . See also

the last section of [12] M 89 .

Lemma 5.4 l-rankformula Let G be a finitely generated pro-p group actingon a pro-p tree T with finite vertex stabilizers such that all edge stabilizersare conjugate to each other. Suppose that there is an edge e ∈ T and a finitesubset V ⊆ DGe such that

• vG = wG implies that v = w for all v, w ∈ V ,

• Gv, v ∈ V generate G.

Then rank(F )− 1 = [G : F ]((|V | − 1)/|Ge| −∑v∈V 1/|Gv|).

Proof: We use induction on [G : F ]. If [G : F ] = 1 there is nothing to prove.Let G1 be the preimage in G of a central subgroup of order p of G/F .

Case 1. G1 ∩Ge is non-trivial.Since for all v ∈ V the edge group Ge is contained in Gv by the hypothesis,Gv centralizes G1 ∩ Ge. But G = 〈Gv | v ∈ V 〉 so G1 ∩ Ge is a centralsubgroup of G of order p. Then, using “bar” to pass to the quotient moduloG1 ∩Ge and the inductive hypothesis, for G we have

rank(F ) − 1 = rank(F ) − 1 = [G : F ]((|V | − 1)/|Ge| −∑v∈V 1/|Gv|) =

([G : F ]/p)(p(|V | − 1)/|Ge| − p(∑v∈V 1/|Gv|)) = [G : F ]((|V | − 1)/|Ge| −∑

v∈V 1/|Gv|) as needed.

Case 2. Ge ∩ G1 = 1. It follows that every torsion element t of G1

is self-centralized. Indeed, by Theorem 3.9 in [14] RZ2 t stabilizes somevertex w, so if g centralizes t, the element t also stabilizes wg. But then byCorollary 3.8 in [14] RZ2 t stabilizes the geodesic [wg,w]. Since, however,Ge ∩ G1 = {1}, the element t cannot stabilize any edge, so wg = w, andtherefore g is a power of t.

Now using this and taking into account that G acts upon the conjugacyclasses of subgroups of order p we may rephrase Lemma 2.6 l-modules-gen (i)as

e-scheiderer G1 =∐v∈V

∐rv∈G/G1Gv

(G1 ∩Gv)rv

q F1, (5)

19

with F1 a free pro-p subgroup of F . Set V1 = {v ∈ V | G1 ∩Gv 6= 1}. Since

G1Gv = FGv for every v ∈ V1 we can rewrite Eq.(5 e-scheiderer ):

G1 =∐v∈V1

∐rv∈G/FGv

(G1 ∩Gv)rv

q F1.

Using the above free decomposition and comparing with Lemma 2.6 l-modules-gen (ii)one finds

|A| =∑v∈V1

|G/FGv| = [G : F ]∑v∈V1

1

|Gv|e-AinLemma (6)

rank(F )− 1 = p rank(F1) + (p− 1)(|A| − 1)− 1 e-rankF (7)

Now p rank(F1) can be computed by observing that passing to the quotientmodulo 〈Tor(G1)〉 and indicating it by “bar” one has rank(F ) = rank(F1),so that using [G : F ] = p[G : F ] the induction hypothesis yields

p rank(F1) = p rank(F )= p|G : F |((|V | − 1)/|Ge| −

∑v∈V 1/|Gv|) + p

= ([G : F ])((|V | − 1)/|Ge| −∑v∈V1

1/|Gv| −∑v∈V \V1

1/|Gv|) + p= [G : F ]/((|V | − 1)/|Ge| −

∑v∈V1

p/|Gv| −∑v∈V \V1

1/|Gv|+ p

(we used Ge ∩ G1 = 1 = Gv ∩ G1 for all v ∈ V \ V1 and |Gv ∩ G1| = pfor all v ∈ V1 to obtain the last equality). Inserting this expression and the

expression for |A| from Eq.(6 e-AinLemma ) into Eq.(7 e-rankF ) yieldsthe claimed formula for rank(F ).

Recall that a pro-p group G acts faithfully on a pro-p tree T if the kernelof the action is trivial and it acts irreducibly if T does not contain a properG-invariant pro-p subtree.

Lemma 5.5 l-exchange1 Let a pro-p group G act faithfully and irreduciblyon a pro-p tree T . Suppose that M is a minimal edge stabilizer and the set ofedges E(TM)G is open in T . Then T ′ := T \ E(TM)G is a subgraph havingeach connected component a pro-p tree.

Contracting each of them results in a pro-p tree T , on which G acts faith-fully and irreducibly. Moreover T = TMG.

Proof: Since T ′ is closed and contains V (T ) it is a subgraph of T . Notealso that m ∈ T ′ if and only if there exists a subgroup L ≤ Gm, an edge

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stabilizer, so that Lg is not contained in M for every g ∈ G. Therefore, sinceM is a minimal edge stabilizer, we can conclude that all edge stabilizers ofedges in TMG are conjugates of M .

Let T be the quotient graph obtained by collapsing each connected com-ponent of T ′ to a vertex. Then T is a G-graph, and, by the Proposition onpage 486 in [25] novosibirsk , it is simply connected and hence a pro-p tree.The collapsing procedure induces a canonical epimorphism which is injectivewhen restricted to E(TM)G.

Let us show that Ge is a conjugate of M for every e ∈ E(T ). Let f ∈E(T ) and u, v be its end points which, by construction, belong to T ′. Fixg ∈ Ge = Gu∩Gv. Then ug and vg belong respectively to the same connectedcomponents as u and v. Since T is a pro-p tree we find that after collapsinge and eg both map to e, and as edges of eG under the collapsing procedureare not identified e = eg must follow. Hence Ge is a conjugate of M indeed.

Suppose T is not irreducible and D were a proper G-invariant subgraph.Since T is obtained by collapsing the connected components of a subgraphthe preimage of every connected subgraph of T in T is connected. Thereforethe preimage of D in T would be a proper connected G-invariant subgraph,a contradiction.

Suppose that g ∈ G acts trivially upon all of T . Then, in particular,eg = e and, as edges of eG under the collapsing procedure are not identified,one must have eg = e, i.e., g ∈ Ge. Therefore the kernel of the action of Gupon T is contained in Ge and so Ge contains a normal subgroup of G which,by Theorem 3.12 [14] RZ2 must act trivially on T , a contradiction. HenceG acts faithfully on T .

Lemma 5.6 l-treetech Let G be a finitely generated pro-p group actingfaithfully and irreducibly on a pro-p tree T with finite vertex stabilizers. Thenthere are a pro-p tree D, an edge e ∈ E(D), a finite subset V ⊆ DGe and afinite subset X ⊆ G such that

(a) G acts faithfully upon D.

(b) All edge stabilizers are pairwise conjugate; in particular, D = DGeG.

(c) vG = wG implies that v = w for all v, w ∈ V .

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(d) X freely generates a free pro-p subgroup H such that for G0 := 〈Gv |v ∈ V 〉 we have G = 〈G0, H〉 and H ∩ (G0)G = {1}.

(e) For all x ∈ XGxe ⊆

⋃v∈V

Gv

Proof: Every maximal finite subgroup of G stabilizes a vertex. If all max-imal finite subgroups of G coincide, then G has a finite normal subgroup,which in light of Theorem 3.13 [14] RZ2 acts trivially on T , contradictingthe faithfulness.

Let e ∈ T be an edge with the stabilizer M := Ge of minimal order.Let Σ denote the set of all finite subgroups L 6= {1} of G that are not

conjugate to a subgroup of Ge. Since G is finitely generated, by Lemma 2.8l-fg-cfg there is a finite subset S of Σ with Σ = {Lg | L ∈ S, g ∈ G}.

Therefore the subset TΣ := {m ∈ T | ∃L ∈ Σ,m ∈ TL}, which is the unionof all subtrees of fixed points TL for subgroups L ∈ Σ can be representedin the form TΣ =

⋃L∈S T

LG and is hence a closed G-invariant subgraphof T . Therefore E(TGe)G = T \ TΣ is open and we can apply Lemma 5.5

l-exchange1 in order to obtain, as a candidate, a simply connected graphD by collapsing the connected components of TΣ on which G acts irreduciblyand faithfully with all edge stabilizers conjugate to M .

Thus D satisfies (a) and (b).We come to proving (c),(d) and (e). Set Q := (G0)G = 〈Gv | v ∈ V (D)〉.By Lemma 5.2 l-action there is finite minimal subset X of generators of

a retract H of G/G0 in F such that DGe ∩DGex is not empty for every x in

X. In fact, as G/Q is free pro-p by Lemma 3.6 in [14] RZ2 , X generates H

freely. Since G is finitely generated, by Lemma 2.8 l-fg-cfg , there exist upto conjugation only finitely many finite subgroups in G. Therefore, by theconstruction of D, there is only a finite subset V of vertices up to translationwith stabilizers that are not conjugates of Ge. To see this one observes that ifvertices v, w are both stabilized by L ∈ Σ, then L stabilizes a geodesic [v, w]

(see Corollary 3.8 in [14] RZ2 ) and so v, w belong to the same connectedcomponent of TΣ.

It follows that G = 〈Gv, H | v ∈ V 〉 = 〈G0, H〉 and G0 ∩ H = {1}, sothat (d) holds. Moreover, since G is pro-p we can reduce V such that no twovertices of it lie in the same orbit. So (c) holds.

22

By construction for every x ∈ X there is a vertex vx ∈ DGe with vxx−1 ∈

DGe . When f is any edge in the geodesic [vx, vxx−1] then Ge = Gf =

Gvx ∩Gx−1

vx, so that, in particular, Gx

e ≤ Gvx . Finally modify V by replacingfor every x ∈ X a vertex v ∈ V by the vertex vx whenever vG = vxG. Thenwe see that (c), (d), and (e) all hold.

Theorem 5.7 t-treeacting Let G be a finitely generated pro-p group actingon a pro-p tree T with finite vertex stabilizers. Then G splits either as a freeamalgamated product or as an HNN-extension over some edge stabilizer.

Proof: By Theorem 3.9 in [14] RZ2 every finite subgroup of G lies in

a vertex stabilizer. Lemma 2.8 l-fg-cfg shows that G has only finitelymany conjugacy classes of finite subgroups and therefore there exists an opensubgroup F of G acting freely on T . So by Proposition 3.5 in [14] RZ2 F

is free pro-p. By Lemma 3.11 in [14] RZ2 there exists a unique minimalG-invariant subtree in T . Replacing T by this subtree we may assume thatthe action of G is irreducible.

We consider G to be a counter example to the theorem with minimalindex [G : F ].

Claim 1: G does not have a finite normal subgroup. In particular, one canassume that G acts on T faithfully and irreducibly.

If G contains a finite normal subgroup, it contains a central one, say C oforder p. By the minimality assumption on |G : F | and as |G/C : FC/C| <|G : F | the quotient group G := G/C satisfies the conclusion of the theorem,i.e. G is either a non-trivial amalgamated free pro-p product G = G1 qH G2

with finite amalgamating subgroup or it is a non-trivial HNN-extension G =HNN (G1, H, t) with finite associated subgroups. Then G is either a non-trivial amalgamated free pro-p product G = G1qHG2 or HNN (G1, H, t) withG1, G2, H being preimages of G1, G2, H in G, respectively, as needed. HenceG does not possess a finite normal subgroup. Since the vertex stabilizers arefinite, the kernel of the action of G upon T is finite, hence it is trivial.

By the preceding lemma, there exist D, G, H, V and G0 satisfying itsconclusions (a)–(e). Note that the stabilizers of vertices in D may well beinfinite.

Claim 2: The free pro-p subgroup H of G must be trivial.

23

Suppose that H 6= 1. Let G = HNN (G0, Ge, X) and λ : G −→ G be theepimorphism given by the universal property. By induction on rank(F ) weshow that λ is an isomorphism.

It suffices to show that the rank of F equals the rank of F := λ−1(F ).If F0 := G0 ∩ F 6= 1 we can factor out the normal closure of F0 in G (and,if necessary, the kernel of the action as well) in order to obtain the quotientgroup G which acts on D/(F0)G and satisfies rank(F ) < rank(F ). Thereforethe induced epimorphism λ : HNN (G0, Ge, X)→ G is an isomorphism, and itis not hard to see that HNN (G0, Ge, X) is isomorphic with G/(F0)G, whereF0 := G0 ∩ F . This shows that the image F of F in G is isomorphic toF /(F0)G. By Lemma 5.3 l-freeedgeaction F is a free product F ∼= F0 q Fand F ∼= F0 q F /(F0)G, so F ∼= F and we are done. Thus we may assume

that G0 is finite. Now applying Lemma 5.4 l-rankformula to G and Gwe deduce that rank(F ) = rank(F ), so λ|F turns out to be an isomorphismcontradicting G being a counter example. This finishes the proof of the claim.

Claim 3: The natural epimorphism λ :∐v∈V GeGv −→ G is an isomor-

phism.

Let us use induction on rank(F ). Since F acts freely on E(D) by Lemma

5.3 l-freeedgeaction , for each v ∈ V the intersection F∩Gv is a free factor ofF , so similarly as in the proof of Claim 2 we can use the induction hypothesis,in order to achieve all Gv to be finite.

Put F = λ−1(F ). Since λ restricted to all Gv is injective, it suffices to

prove that λ|F is an isomorphism. But by applying Lemma 5.4 l-rankformula

to G and G we get that F and F have the same rank and therefore λ is anisomorphism. The result follows.

Claim 3 shows that G is not a counter-example, a final contradiction.

Now we are ready to prove Theorem 1.2 t-stallings .

Theorem 5.8 t-stallings1 Let G be a finitely generated virtually free pro-pgroup. Then G is either a non-trivial amalgamated free pro-p product withfinite amalgamating subgroup or it is a non-trivial HNN-extension with finiteassociated subgroups.

Proof: By Theorem 3.3 t-HNN G embeds into a group G = E×G/F suchthat every finite subgroup of G is conjugate to a subgroup of G/F and E isfree pro-p.

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By Theorem 10.1 t-models G is a permutational extension of E and soby Remark 3.5 rem-perm-ext an HNN-group HNN (G/F,G,Γ) where the

base group G/F and the associated groups in G are all finite. Thus G actson a pro-p tree T such that T/G is a bouquet and all vertex stabilizers are

finite (see [14] RZ2 , page 89 for the situation of a single loop).Consider the restriction of this action to G. Now we apply Theorem 5.7

t-treeacting to conclude the result.

Theorem 5.9 t-fund A finitely generated virtually free pro-p group G is

isomorphic to the fundamental pro-p group Π1(G,Γ, T ) of a finite graph offinite p-groups.

Proof: To see ‘if’ choose an open normal subgroup U of G that intersectstrivially with the vertex groups. Then by Section 3 in [21] ZM 90 G acts onthe standard pro-p tree S with finite vertex stabilizers and so U acts freelyon S. Then by Theorem 3.4 in [14] RZ2 U is free pro-p.

For the ‘only if’-part use induction on the rank of a maximal normalfree pro-p subgroup F of G. By Theorem 5.8 t-stallings1 G splits as apro-p amalgamated product G = G1 qK G2 or pro-p HNN-extension G =HNN (G1, K, t) over a finite subgroup K. Moreover, we are free to chooseK up to conjugation in G1. Then every factor (respectively the base group)satisfies the induction hypothesis and so is the fundamental group of a finitegraph of finite p-groups. By Theorem 3.10 in [22] Z-M 89b K is conjugateto some vertex group of G1 and so we may assume that K is contained ina vertex group of G1. Now in the case of an amalgamated product there isg2 ∈ G2 such that Kg2 is in a vertex group of G2, so G admits a decompositionG = Gg2

1 qKg2 G2. Thus in both cases G becomes the fundamental group ofa finite graph of finite finite p-groups.

Example 5.10 t-4.3 Let A and B be groups of order 2 and G0 = 〈A×B, t |tAt−1 = B〉 be a pro-2 HNN extension of A×B with associated subgroups Aand B. Note that G0 admits an automorphism of order 2 that swaps A andB and inverts t. Let G = G0×C be the holomorph. Set H0 = 〈Tor(G0)〉 andH = H0×C. Since G0 is virtually free pro-2, so is G and therefore so is H.The main result in [8] HZ archiv shows that H does not decompose as thefundamental group of a profinite graph of finite 2-groups.

25

6 Automorphisms

s-autos

The following theorem is a consequence of Theorem 3.3 t-HNN :

Theorem 6.1 t-autoembeding Let Fn be a free pro-p group of finite rank nand P a finite p-group of automorphisms of F . Then there is an embeddingof holomorphs Fn×P −→ Fm×P such that P permutes the elements of somebasis of a free pro-p group Fm.

For a finite set X the canonical embedding of the discrete free groupΦ(X) into its pro-p-completion F (X) induces an embedding Aut(Φ(X)) ≤Aut(F (X)).

Theorem 6.2 t-CC Let F = F (X) be a finitely generated free pro-p groupand Φ = Φ(X) be a dense abstract free subgroup of F on the same set ofgenerators. Suppose that A ≤ Aut(F ) is a finite p-group. Then there existsan automorphism β ∈ Aut(F ) such that the conjugate Aβ is contained inAut(Φ).

Proof: Indentifying F with its group of inner automorphisms, we may con-sider the holomorph G := F×A as a subgroup of Aut(F ). Since G is a finitely

generated virtually free pro-p group, we may use Theorem 5.9 t-fund in or-der to present G as the fundamental group of a finite graph (G,Γ) of finitep-groups. Therefore G acts faithfully with finite stabilizers on a pro-p tree.By Theorem 5.6 of [21] ZM 90 every finite subgroup of G is conjugate toa subgroup of a vertex group, so there exists β0 ∈ G with Aβ0 ∈ G(v) forsome v ∈ V (Γ). Let π1(G,Γ) be the abstract fundamental group of the same

graph of groups (see, e.g., [2] D 80 ), and set Φ0 := π1(G,Γ) ∩ F . Choose abasis Y of Φ0. Then Y is a basis of F (X), thus there exists α ∈ Aut(F (X))sending X bijectively to Y . For β := β0α

−1, Aβ ≤ Aut(Φ).

Theorem 6.3 t-autos Let F be a free pro-p group of rank n.

(i) The embedding Aut(Φ) ≤ Aut(F ) induces a surjection between theconjugacy classes of finite p-subgroups of Aut(Φ) and Aut(F ).

26

(ii) The Aut(F )-conjugacy classes of finite subgroups of Aut(F ) of ordercoprime to p are in one-to-one correspondence with Aut(F/Φ(F ))-conjugacy classes of finite subgroups of Aut(F/Φ(F )) ∼= GLn(Fp) oforder coprime to p.

Proof: Statement (i) is a consequence of Theorem 6.2 t-CC .We begin the proof of (ii) by defining a homomorphism λ : Aut(F ) →

Aut(F/Φ(F )) setting

λ(α)(fΦ(F )/Φ(F )) := α(f)Φ(F )/Φ(F ).

Lemma 4.5.5 [13] RZ1 2000 shows that the kernel K := kerλ is a pro-p group. Moreover, λ is an epimorphism, since every automorphism α ∈Aut(F/Φ(F )) can be lifted to an automorphism of F .

Let us first show that every p′ subgroup Q of Aut(F/Φ(F )) is of the formQ = λ(Q0) for a suitable p′-subgroup Q0 of Aut(F ). Indeed, λ−1(Q) containsthe normal p-Sylow subgroup K and therefore by the profinite version of theSchur-Zassenhaus theorem 2.3.15 [13] RZ1 2000 , λ−1(Q) is a split extensionof the pro-p group K by a p′-group Q0, i.e., λ−1(Q) = K×Q0 and so Q =λ(Q0), as desired.

Next suppose that A and B are p′-subgroups of Aut(F ) so that λ(A)and λ(B) are conjugate in Aut(F/Φ(F )). Then there exists g ∈ F so thatAgK = BK. Now K is a closed normal p-Sylow subgroup of BK andK ∩Ag = K ∩B = {1} shows that Ag and B are complements of K in BK.

Therefore, again by Theorem 2.3.15 [13] RZ1 2000 , they are conjugates inBK. Hence A and B are conjugate in G.

References

[1] Z.M. Chatzidakis, Some remarks on profinite HNN extensions, Isr.J. Math. 85, No.1-3, (1994) 11-18.

[2] W. Dicks, Groups, Trees and Projective Modules, Springer 1980.

[3] A. Heller and I. Reiner, Representations of cyclic groups in rings ofintegers. I, Ann. of Math. (2) 76 1962 73–92

[4] W. Herfort, P.A. Zalesskii and L. Ribes, p - Extensions of free pro-pgroups, Forum Math. 11 (1998), 49–61.

27

[5] W. Herfort and P.A. Zalesskii, Cyclic Extensions of free pro-p groups,Journal of Algebra, 216 (1999) 511–547.

[6] W. Herfort and P.A. Zalesskii, Virtually free pro-p groups whose tor-sion elements have finite centralizers, Bull. London Math. Soc. 200840 (2008) 929–936.

[7] W. Herfort and P.A. Zalesskii, Profinite HNN-constructions, J.Group Theory 10(6) (2007) 799–809

[8] W. Herfort and P.A. Zalesskii, A virtually free pro-p group need notbe the fundamental group of a profinite graph of finite groups, Arch.d. Math. (to appear)

[9] A. Karrass, A. Pietrovski and D. Solitar, Finite and infinite cyclicextensions of free groups, J.Australian Math.Soc. 16 (1973) 458–466.

[10] D. Kochloukova P.A. Zalesskii, Automorphisms of pro-p groups of fi-nite virtual cohomological dimension, Q. J. Math. 58(1) (2007) 47–51.

[11] A.A. Korenev, Pro-p groups with a finite number of ends, Mat. Za-metki 76 (2004), no. 4, 531–538; translation in Math. Notes 76(2004), no. 3-4, 490–496.

[12] O.V. Mel’nikov, Subgroups and Homology of Free Products of Profi-nite Groups, Math. USSR Izvestiya, 34, 1, (1990), 97-119.

[13] L. Ribes and P.A. Zalesskii, Profinite Groups, Springer 2000.

[14] L. Ribes and P.A. Zalesskii, Pro-p Trees and Applications, (2000),Chapter, Ser. Progress in Mathematics, Birkhauser Boston (2000),Ed. A. Shalev, D. Segal.

[15] L. Roman’kov, Infinite generation of automorphism groups of freepro-p groups, Siberian Mathematical Journal 34, (1993) 727-732

[16] C. Scheiderer, The structure of some virtually free pro-p groups,Proc. Amer. Math. Soc. 127 (1999) 695-700.

[17] J.-P. Serre, Sur la dimension cohomologique des groupes profinis,Topology 3, (1965) 413-420.

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[18] J.R. Stallings, Applications of categorical algebra. Proc. Sympos.Pure Math., Vol. XVII. American Mathematical Society, 124 (1970)124-129.

[19] J.S. Wilson, Profinite Groups, London Math.Soc. Monographs(Clarendon Press, Oxford, 1998)

[20] P.A. Zalesskii, A geometric characterization of free formations ofprofinite groups, Sib. Math. J. 30, No.2, 227-235 (1989), (transla-tion from Sib. Mat. Zh. 30, No.2(174), 73-84 (1989))

[21] P.A. Zalesskii and O.V. Mel’nikov, Fundamental Groups of Graphs ofProfinite Groups, Algebra i Analiz 1 (1989); translated in: LeningradMath. J. 1 (1990), 921-940.

[22] P.A. Zalesskii and O.V. Mel’nikov, Subgroups of profinite groups act-ing on trees, Math. USSR Sbornik 63 (1989) 405-424.

[23] P.A. Zalesskii, Normal subgroups of free constructions of profinitegroups and the congruence kernel in the case of positive characteris-tic, Izv. Ross. Akad. Nauk 59, 59-76 (1995); translated in: Izv. Russ.Acad. Sciences, Ser Math. (1996)

[24] P.A. Zalesskii, Virtually projective groups, J. fur die Reine und Ange-wandte Mathematik (Crelle’s Journal) 572 (2004) 97-110.

[25] P.A. Zalesskii, Open Subgroups of Free Profinite Products, Proceed-ings of the International Conference on Algebra, Part 1 (Novosibirsk,1989), 473–491, Contemp. Math., 131, Part 1, Amer. Math. Soc.,Providence, RI, 1992.

University of Technology at Vienna, Austria, w.herfort@tuwien.ac.atUniversity of Brasilia, Brasilia-DF, Brazil, pz@mat.unb.br

29

7 Strategy

Note 126.1.The elementary abelian case seems to work. Perhaps myproof is somehow lengthy – however I think one needs toinclude a proof for lifting finite subgroups – which I did.There are still difficulties in the general case – mainlywhen necessary to lift finite subgroups or torsion.

Notation 7.1 n-NH For a virtually free pro-p group G and H / G letNH denote the subgroup generated by all CG(x) with x a nontrivial torsionelement in H. Observe that NH = 〈CG(x) | 1 6= x ∈ H, xp = 1〉.

We collect several properties of a permutational extension G.

Remark 7.2 rem-permext-lifting When G = HNN (K,Ai, Zi, F ) is a per-

mutational extensionas im Remark 3.5 rem-perm-ext with F := (Zi | i ∈I)G and there are given a finite subgroup T / K and s ∈ socle(K) and anelement t ∈ G of order p then the following facts hold:

(a) G = F×K and F is free pro-p;

(b) Every finite subgroup of G is F -conjugate to a subgroup of K;

(c) Write “bar” for passing to the quotient modulo (CF (s))F . Then

1. NG(t) = NG(t);

2. NH = NH ;

3. Every finite subgroup of G is F -conjugate to a subgroup ofK;

(d) Set H := FT , assume that H /G and write “bar” for passing to thequotient modulo NH . Then

1. NG(t) = NG(t);

30

2. Every finite subgroup of G is F -conjugate to a subgroup ofK;

Notation 7.3 n-FK Until the end of this section we analyze a semidirectproduct G = F×K in which every finite subgroup is conjugate to a subgroupof K. Some of the properties mentioned above will be established underadditional assumptions on G. Then we shall fix s ∈ socle(K), an elementt ∈ G of order p, and, F ≤ H / G.

Lemma 7.4 l-conjugacy1 Let G = F×K be a semidirect product, with K afinite p-group and F free pro-p. The following properties are equivalent:

(a) Every finite subgroup of G is conjugate to a subgroup of K;

(b) For every finite subgroup T ≤ K, every torsion element of NG(T ) isconjugate by an element in CF (T ) to an element in NK(T ).

Proof: “(a) ⇒ (b)” Let T ≤ K and x ∈ Tor(NG(T )). Then 〈x, T 〉 isfinite, and so is conjugate by an element f ∈ F to a subgroup of K. Inparticular, for all t ∈ T , we have t−1tf ∈ K ∩ F = {1}, so that f ∈ CF (T )and xf ∈ NK(T ).

“(b)⇒ (a)” Suppose that |K| is minimal such that the conclusion fails tohold for a finite subgroup R of G. Let F ≤ H be any maximal open subgroupof G. By the minimality assumption on |K| the subgroup T := H∩R satisfiesthe conclusion and so we can assume that T ≤ K. Fix any element x ∈ R\H– we have then R = T 〈x〉. Then x ∈ Tor(NG(T )) and so there is f ∈ CF (T )with xf ∈ K. Then Rf ≤ K follows, a contradiction.

Lemma 7.5 l-cpcp Let G be as in Notation 7.3 n-FK and K = 〈s〉 ×〈t〉 ∼= Cp×Cp. Then F admits a free decomposition into 〈t〉-invariant factorsF = CF (K)qCtqCsqPt, such that CF (t) = CF (K)qCt, CF (s) = CF (K)qCsand t acts freely on Cs and Pt.

In particular, (CF (s))F ∩CF (t) = CF (K), and, writing “bar” for passingto the quotient modulo (CF (s))F , we have CF (t) = CF (t).

Proof: There are free decompositions CF (t) = CF (K)q Ct and

e-Ft 〈t〉F = (〈t〉 × CF (t))q Ft (8)

31

with Ft free pro-p. By Lemma 2.9 l-conjugacy every finite subgroup of

CG(s) = (CF (s)×〈s〉)×〈t〉 has finite subgroups conjugate to subgroups of K;the same observation holds for its subgroup CF (s)×〈t〉. Therefore the latter

has a free decomposition CF (s)×〈t〉 = (CF (K) × 〈t〉) q F1. Eq.(8 e-Ft )gives rise to decompose M = F/[F, F ] as a 〈t〉-permutational Heller-Reinermodule M = M1 ⊕Mp. The image, say L, of the 〈t〉-invariant free factor

Cs :=∐p−1i=0 F

ti

1 in CF (s) is a free 〈t〉-module and so, by Corollary 2.4 c-subthe submodule Mp can be chosen to contain L as a direct summand. Thelatter gives rise to a free decomposition of F into 〈t〉-invariant free factorsF = CF (K)q Ct q Cs q Pt, so that 〈t〉 acts freely on Cs and Pt.

The equality (CF (s))F ∩ CF (t) = CF (K) is a consequence of this de-composition of F and the last statement of the lemma follows from the justestablished equality and an application of the second isomorphism theorem.

Lemma 7.6 l-CFt Let G, H and t be as in Notation 7.3 n-FK ands ∈ socle(K). Indicate by “bar” passing to the quotient modulo (CF (s))F .Then

(i) Every finite subgroup of G is conjugate to a subgroup of K.

(ii) NH = NH ;

Proof:Suppose that (i) is false and G is a counter-example with |K| minimal.

Then certainly CF (s) 6= {1}. Let T ≤ K arbitrary and x a torsion element inNG(T ). We shall show that x is conjugate to an element in NK(T ) by some

element in CF (T ). Lemma 7.4 l-conjugacy1 then will yield a contradiction.

By the minimality assumption on |K| we must have NK(T ) = K. Wecannot have s ∈ T since (i) implies that CF (s) = {1} and so CF (T ) = {1};hence NG(T ) must be finite and so it is contained in K, a contradiction.

So there exists 1 6= t ∈ T∩socle(K). Lemma 7.5 l-cpcp implies NG(t) =

NG(t) where t is the unique element in K with t(CF (s))F/(CF (s))F = t.The same lemma shows that (CF (s))F ∩ CF (t) = (CF (s, t))CF (t), giving

rise to an isomorphism ψ : NG(t) → NG(t)/(CF (s, t))CF (t). By Lemma 2.9

l-conjugacy every finite subgroup of NG(t) is conjugate to a subgroup of

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K. This observation also holds in NG(t)/〈t〉 and so, by the minimality as-sumption on |K|, we can conclude that every finite subgroup of NG(t)/〈t〉 ×(CF (s, t))CF (t) is conjugate to a subgroup of K/〈t〉 by an element f in CF (t).Therefore every finite subgroup in NG(t) is conjugate by an element in CF (t)to a subgroup of K – we have used the isomorphism ψ as well. Now Lemma7.4 l-conjugacy1 implies that for every finite subgroup L of NG(t) every tor-

sion element y in NG(t) is conjugate to an element in NK(L) by an elementf ∈ CF (L). In particular this statement holds with y := x and L := T , as has

been claimed at the beginning of the proof. Now Lemma 7.4 l-conjugacy1

implies (i) to hold for G, a contradiction.

(i) is a consequence of Lemma 7.5 l-cpcp sinceNH = 〈CF (t) | 1 6= t ∈ H〉 =

〈CF (t) | 1 6= t ∈ H〉 and, using (i), we infer that the latter equals 〈CF (t) |1 6= t ∈ H〉 = NH .

Corollary 7.7 c-elab Suppose that K is elementary abelian and every fi-nite subgroup of G is conjugate to a subgroup of K. Fix a maximal subgroupH of G containing F . Then every finite subgroup of G/NH is conjugateto a subgroup of KNH/NH and CG/NH

(tNH/NH) = CG(t)NH/NH for everytorsion element t ∈ K.

Proof: Assume that the Corollary is false. Then there is a counter-exampleG so that TH := {1 6= x ∈ H ∩K | CF (x) 6= {1}} has minimal cardinality.There must exist s ∈ TH , else NH = {1} and then G cannot be a counter-example. Let “bar” denote passing to the quotient modulo (CF (s))F . Lemma

7.6 l-CFt implies that

e-NHol CF (t) = CF (t), NH = NH , (9)

and that every finite subgroup of G is conjugate to a subgroup of K. Since THhas fewer elements than TH the minimality assumption on |TH | implies thatCF (t)NH/NH = CF/N

H(tNH/NH), and that every finite subgroup of G/NH

is conjugate to a subgroup of KNH/NH . Now, the second isomorphism

theorem and Eq.(9 e-NHol ) imply the statements of the lemma to hold forG, a contradiction.

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Note 2wh 16.2.after discussions in Milano

Lemma 7.8 l-auto-finite-edge-stab Let the pro-p group G act on a pro-ptree T with a single conjugacy class of finite edge stabilizers. Assume thatno vertex stabilizer coincides with an edge stabilizer. Then for any automor-phism α ∈ Aut(G) and edge e the image α(Ge) is an edge stabilizer.

Proof: Let u and v be the endpoints of e. Then α(Ge) = α(Gu ∩ Gv) =α(Gu)∩α(Gv). There are vertices u′, v′ in T with α(Gu) ≤ Gu′ and α(Gv) ≤Gv′ . By our assumptions we cannot have Gu = Gv and so Gu′ ∩Gv′ can onlybe an edge stabilizer, say Gf for some edge f . Hence α(Ge) = Gf as claimed.

Corollary 7.9 c-auto-perm-ext When G = HNN (K,Ai, Zi, I) then, for

every automorphism α of G and every j ∈ I, there exists gj ∈ G with α(Aj) =Agj

j .

Proof: We can assume Aj 6= {1} and fix z ∈ Zj. Then, Putting Z ′i := Zifor i 6= j and Z ′j := Zj \ {z} we can write G = HNN (H,Aj, {j}, {z}) withH := HNN (K,Ai, Z

′i, i ∈ I \ {j}. Now G acts on the standard graph and

satisfies the assumptions of Lemma 7.8 l-auto-finite-edge-stab . Since all

edge stabilizers are conjugate we find that α(Aj) = Agj

j for a suitable elementgj ∈ G as claimed.

Lemma 7.10 l-ivstrengthen-new-new Let G be the semidirect product ofa finitely generated free pro-p group F with a finite p-group K. Supposethat there is a maximal open subgroup H of G containing F and H is apermutational extension. Assume that at least one of the following hold:

(a) Every finite subgroup of G is conjugate to a subgroup of K.

(b) M := F/[F, F ] is a K-permutation module w.r.t. the induced repre-sentation of K acting by conjugation on F .

34

Letting “bar” indicate passing to the quotient modulo NH , the following state-ments hold true:

(i) F is free pro-p;

(ii) Every torsion element of G is conjugate to an element in K andCF (t) = CF (t) for every element t in G of order p.

Proof: (i) is true since by assumption H is a permutational extension.

(ii) Put U := H ∩ K. Suppose first that a torsion element t in G doesnot lift. Certainly t 6∈ H. Let us first show that tp = 1. Indeed, if tp 6= 1then tp ∈ H = F×U and since H satisfies the statements of the lemma thereare a torsion element y ∈ K and and f ∈ F with tp = yf . There is no lossin generality to assume tp = y. Since G = F×K there exist k ∈ K andφ ∈ F with φ k = t. From k

p= tp we deduce that 〈k, t〉 ≤ CG(tp) holds and

therefore φ ∈ CF (tp) = {1}; hence t = k, a contradiction. Thus tp = 1.

Fix k ∈ K and φ ∈ F with t = φ k. Then kp = 1 and k ∈ K \ H.By assumption H = HNN (U,Ai, Zi, i ∈ I) is a permutational extension.Conjugation by k induces an automorphism on H and by Corollary 7.9c-auto-perm-ext for every i ∈ I with Ai 6= {1} there exist j(i) ∈ I and

hi ∈ H with Aki = Ahi

j(i). The induced permutation action of 〈k〉 on I admitsa decomposition I = I1 ∪ Ip where I1 contains all the fixed points and Ipconsists of the cycles of length p.

We next intend to exchange the subgroups Ai 6= {1} and some of the Ziin such a way that Z :=

⋃i∈I,Ai 6={1}(Ai\U) × Zi) becomes 〈k〉-invariant and

so that H is still a permutational extension.Fix a representative set Jp of Ip/〈k〉 in Ip and replace Zikj by Zkj

i fori ∈ Ip and j = 0, . . . , p− 1. Now all Zi are 〈k〉-invariant for i ∈ Ip.

Next suppose i ∈ I1. Then kh−1i ∈ NG(Ai). Writing hi = fiui with

fi ∈ F and ui ∈ U = H ∩ K and considering Akh−1

ii = Ai modulo F shows

that ku−1i ∈ NK(Ai) holds.

For every i ∈ I1 set Bi := Auii and for i ∈ Ip set Bi := Ai. Then the

universal property of the permutational extension, considered as an HNN-extension (see Definition 3.1 d-HNN-ext and Remark 3.5 rem-perm-ext )

shows that H = HNN (K,Bi, Zi, I).The set (Ai\U) × Zi agrees with (Bi\U) × Zi – as can be seen from the

fact that air = a′uii s has a solution (a′i, s) ∈ Ai × U for every given pair

35

(ai, r) ∈ Ai × U . As a result we find that k by conjugation permutes theelements of the (Ai\U) × Zi. We shall find it convenient to rename the Bi

into Ai and to keep in mind that Z =⋃i∈I,Ai 6={1}(Ai\U)×Zi is 〈k〉-invariant,

as desired.Note that L := 〈Z〉 is 〈k〉-invariant and that F = L q FH for a suitable

free pro-p subgroup FH of F . Moreover, since k permutes the generators ofL the subgroup L×〈k〉 is a permutational extension and so, by Lemma 2.6

l-modules-gen , it is isomorphic to (〈k〉 × CL(k)) q F0k for a suitable freepro-p subgroup F0k. Therefore

e-FH F = (CL(k)q FL)q FH (10)

where 〈k〉 acts freely on FL =∐p−1j=0 F

kj

0k .On the other hand, 〈k〉 acts on F . By our assumptions 〈k〉 induces 〈k〉 per-

mutation representation onM = F/[F, F ] and by Lemma 2.6 l-modules-gen

we can conclude that F×〈k〉 has only a single conjugacy class of subgroupsof order p. Since CF (k) = CL(k)q F1, we can freely decompose

e-Fk F = (CL(k)q F1)q Fk, (11)

where 〈k〉 acts freely on Fk.Using these decompositions it is our intention to find a free factor F0H

on which 〈k〉 acts freely so that

e-FkH F = F1 q FL q CL(k)q F0H . (12)

Having manufactured such a decomposition we can see that F1 q F0H is U -invariant and since the normal closure of L = FL q CL(k) in F agrees withNH it shows that F/NH×〈k〉 can contain only a single conjugacy class ofsubgroups of order p. Therfore t would be a conjugate of k, a contradiction.Similarly the second statement – CF (k)NH/NH = CF/NH

(kNH/NH) for everyk of order p in G – is proved.

So, for finishing the proof of the lemma, it remains to achieve a decom-position as in Eq.(12 e-FkH ).

Letting – temporarely – “bar” denote passing to the quotient modulo(CL(k))F [F, F ] the decompositions in Eq.(10 e-FH ) and Eq.(11 e-Fk )show that F is a 〈k〉-permutation module possessing a Heller-Reiner decom-position

F = M1 ⊕Mp

36

with M1 = F 1 and Mp =⊕p−1

j=0 Fkj

0k. Now using Corollary 2.4 c-sub weexchange Mp in this decomposition so that it contains the free 〈k〉-submoduleFL. Lifting its generators to F/(CL(k))F and finally to F yields the desired

free decomposition in Eq.(12 e-FkH ).

We come to the main result of this section comprising Theorem 1.3t-permlift .

Theorem 7.11 t-characterizations Let G be the semidirect product of afinitely generated free pro-p group F by a finite p-group K. The followingproperties are equivalent:

(i) G is a permutational extension.

(ii) Every finite subgroup of G is conjugate to a subgroup of K.

(iii) K by conjugation turns M := F/[F, F ] into a K-permutation module.

Proof: (i) implies both, (ii) and (iii), as has been remarked in Remark 3.5rem-perm-ext .

Suppose now that either (ii) or (iii) holds and that (i) is false. Then thereis an example G with |K| minimal.

For every maximal open subgroup H of G that contains F either of (ii)or (iii) holds and by the minimality assumption on |K| we find that H isa permutational extension. If for each such H the intersection K ∩ H actsfreely on F then K acts freely and then by Prop. 14 [6] paper3 we find

G =(∐

j∈J Aj)q F0 for finite p-groups Aj, a finite set J and a free pro-

p group F0. Either of the assumptions (ii) or (iii) implies that J cannotcontain more than one element and so G = HNN (K, {1}, Z) for Z a freeset of generators of F0 and the index set I has only one index for Ai = {1}.Since G is supposed to be a counter-example, we have a contradiction.

Thus there must be an open maximal subgroup H of G containing F notacting freely on F . HenceNH 6= {1}. Now Lemma 7.10 l-ivstrengthen-new-new

implies thatG/NH satisfies the assumptions of Theorem 12.6 t-modelsstep1 .

Therefore G/NH is a permutational extension. Using this and again Lemma

7.10 l-ivstrengthen-new-new shows that the assumptions of the criterion

Theorem 7.13 t-models-new hold. Hence G is a permutational extension,a contradiction.

37

Note 3wh 4.2.Here is a variant of the induction engine: with P =Φ(K)F finite subgroups of G/NP lift. To my mind itshould replace the proof of Subclaim 1 of Theorem 10.1t-models

Recall Notation 7.1 n-NH .

Lemma 7.12 l-ivstrengthen-sgrps Let G be a semidirect product of a finitelygenerated free pro-p group F with a finite p-group K such that every fi-nite subgroup of G is conjugate to a subgroup of K and suppose that everyproper open subgroup H of G containing F is a permutational extension. SetP := FΦ(K). Let “bar” indicate passing to the quotient modulo NP .

The following statements hold true:

(i) Every finite subgroup of G is conjugate to a subgroup of K;

(ii) F is free pro-p;

(iii) CF (t) = CF (t) for every element t in G of order p.

Proof: (i) Suppose the statement is false. We consider a counter-exampleso that the cardinality of the set {1 6= k ∈ K | CF (k) 6= {1}} is minimal.

Lemma 7.4 l-conjugacy1 implies that elements in the socle of Φ(K) actfixed point free.

Since G is a counter-example, in light of Lemma 7.4 l-conjugacy1 , there

exists a subgroup T ≤ K and a torsion element x ∈ NG(T ) which cannot beconjugated by an element of CF (T ) to an element in K. If 〈x〉TP < G then〈x〉TP is a proper open normal subgroup and so contained in a maximal opensubgroup, say H, where H, by assumption, is a maximal open subgroup ofG. By assumption H is a permutational extension and so is H. Thereforethere must be f ∈ CF so that xf ∈ K, a contradiction.

Thus 〈x〉TP = G. Considering this equality modulo F we find 〈x〉T ∼= K.Hence there exists an element s ∈ socle(K) belonging to the socle of 〈x〉Tas well. Therefore x ∈ NG(s). There is k ∈ K with xk−1 ∈ F . But thenxk−1 ∈ CF (s) = {1} so that x ∈ K, a contradiction.

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(ii) holds since P is a permutational extension. (iii) is a consequence of(i).

39

7.1 Final proof in (old) section 4

Note 4wh 26.1.

In this form Theorem 10.1 t-models becomes a tool.An inspection of its proof shows that everything beforethe Claim and the Claim itself can be omitted.

Theorem 7.13 t-models-new Let G = F×K be a finitely generated pro-pgroup with F free pro-p and K a finite p-group. Suppose there is U <p K sothat H := FU satisfies the following conditions:

a) H and G/NH are permutational extensionsb) CF/NH

(t) = CF (t)NH/NH holds for all t ∈ K of order p.Then G is a permutational extension.

Proof: The proof of Theorem 10.1 t-models comes here in a modifiedform.

Theorem 7.14 t-structure Every finitely generated virtually free pro-p groupG = F×K with all finite subgroups conjugates of subgroups of K is a per-mutational extension.

Proof: Suppose the theorem is false. There is a counter-example G = F×Kwith minimal K and minimal rank of F .

We claim that for every maximal subgroup U of K there exists t ∈ Kwith K = L×〈t〉.

If there is H containing all elements of K of order p then NH = NK .Then NH 6= {1} since otherwise K acts by conjugation freely on F andtherefore Prop. 14 [6] paper3 implies that G = K q F0 with F0 free pro-p.

Then G is a permutational extension, a contradiction. So NH 6= {1}. Since|K| has been chosen minimal the subgroup H is a permutational extension.

Therefore Remark 7.2 rem-permext-lifting shows that every torsion element

in H acts freely by conjugation on F/NH . Since K \ H does not containelements of order p, in the quotient G/NH , one cannot find elements oforder p outside H/NH . Hence KNH/NH acts freely on F/NH . Therefore

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Prop. 14 [6] paper3 implies that G/NH = (∐i∈I Ai) q F0 for a finite set

I of finite p-groups Ai and F0 free pro-p. Let us show that I can consistof a single element only. Indeed, taking into account that there is a singleconjugacy class of maximal finite subgroups in H/NH , we find H/NH = ((H∩K)NH/NH)qF1 for a suitable free pro-p subgroup F1. On the other had, by

the Kurosh subgroup theorem 9.1.9 [13] RZ1 2000 we can see that H/NH =(∐i∈I

(∐τi∈Ji

(Ai ∩H/NH)τi))q F2 with F2 free pro-p. Comparison of the

two free decompositions of H/NH shows that Ai∩H/NH must be trivial withthe only exception when Ai ∩H/NH is conjugate to (H ∩K)NH/NH . SinceAi \H/NH does not contain elements of order p the group G/NH is indeed ofthe form G/NH = KqF1. Then G/NH is a permutational extension. Since His a permutational extension and since CG/NH

(tNH/NH) = CG(t)NH/NH ={1} as a consequence of of the assumption that NH = NG, we find the

premises of Theorem 7.13 t-models-new to hold for U := H ∩K. Hence Gis a permutational extension, a contradiction.

We next claim that K cannot be elementary abelian. If K were cyclic,Prop. 9 [4] HRZ 99 shows that G = K q F0 with F0 free pro-p. Such G isa permutational extension, a contradiction.

Hence there is a maximal open subgroup H of G containing F prop-erly. Among all such examples G and H we fix one with TH = {1 6=x ∈ K | CF (x) 6= {1}}. We now check that the premises of Theorem 7.13

t-models-new hold. Then G will turn out to be a permutational extension,a final contradiction.

By the minimality assumption on |K| we can conclude that H is a per-

mutational extension. Since K is elementary abelian, Corollary 7.7 c-elabimplies that in G/NH all finite subgroups are conjugate to subgroups ofKNH/NH . Therefore, since TK/NH

has fewer elements than TH , we can con-clude that G/NH is a permutational extension. The only condition yet to beverified is CF/NH

(tNH/NH) = CF (t)NH/NH for arbitrary t ∈ K. The latter

also follows from Corollary 7.7 c-elab .

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Note 5Still something missing. What we know up to now isonly:

1. K is not elementary abelian;

2. G = H×〈t〉 for every maximal subgroup H ≥ Fof G and some t of order p;

3. it seems that TH is an interesting quantity for in-ducting. Sometimes better than the rank of F .

4. The very strong Theorem 12.6 t-modelsstep1 hasnot yet been used. At do not see how.

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8 Structure theorem – Example(s)

Proposition 8.1 p-fsc Consider the pro-2 HNN-extension G = HNN (K,A, t)

with K := 〈a〉 × 〈b〉 where a2 = b2 = 1 and A = 〈a〉, At = 〈b〉. The followingproperties hold:

(i) Every finite subgroup of G is conjugate to a subgroup of K;

(ii) G is not of the form F×K with F free pro-2 and K finite;

(iii) G is not a permutational extension.

Proof: (i) is a consequence of Theorem 4.1 (c) [14] RZ2 . Suppose nextthat (ii) were false. Then G = F×K and, using (i), we can assume thatK = 〈a, b〉. Then [G,G] ∩ K = {1}. On the other hand we have [a, t] =at−1at = ab ∈ [G,G] ∩ K, a contradiction. Now (iii) follows from (ii) and

Definition 3.4 d-perm-ext of permutational extension.

9 Lifting centralizers – Examples

The discussion to follow concerns Lemma 4.1 l-ivstrengthen-new and The-

orem 4.4 t-permutational . Proposition 9.2 p-cdl-conj below shows that itis not enough to stipulate that every torsion element in F×K is conjugate toan element in K. It also shows that centralizers in F do not necessarily liftwhen we factor the F -centralizers of elements in a proper subgroup of K – anobstruction for proving (ii) in Lemma 4.1 l-ivstrengthen-new (the situation

in Claim 2 where K = Cp×Cp). On the other hand in both examples torsionlifts.

Proposition 9.2 p-cdl-conj contains a variant of the following examplefrom 2007.

Proposition 9.1 p-cdl Let G = 〈a, b, t | ap = bp = (a, b) = (at, b) = 1〉.Set K := 〈a, b〉. The following statements hold:

(i) K ∼= Cp × Cp.

(ii) G is an HNN-extension of the form G = HNN (H, 〈a〉, 〈c〉, t) whereH := (〈a〉 × 〈b〉)q〈b〉 (〈b〉 × 〈c〉) and c = at.

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(iii) The normal closure F := (t)G = F (x0, x1, . . . , xp−1, z) is a free pro-p

group of rank p + 1 where, for i = 0, . . . , p − 1 we put xi := tbi

andz := t−1ta.

K acts as a group of automorphisms as follows: xai = xiz, za = z−1,xbi = xi+1 (mod p), z

b = z.

(iv) G is the pro-p fundamental group of the following graph of groups:

〈a, b〉〈b〉

((

〈a〉66〈b, c〉 Here we let the boundary maps identify along the

upper edge the two b’s and along the lower edge a with c. Whenchoosing the upper edge together with the end points as a spanningtree, we find that the lower edge yields the relation c = at.

(v) We have G = F×K with K := 〈a, b〉 so that n(G) = 2.

(vi) The element ab is not conjugate to atb.

(vii) For L := F×〈b〉 we have CF (b) = 〈z〉 with z = [a, t] and L =(〈b〉 × 〈z〉)q 〈t〉. Moreover G/NL

∼= K × 〈t〉.In particular, for k ∈ K\〈b〉 we have CF/NL

(kNL/NL) > CF (k)NL/NL.

We have, however, Tor(G/NL) = Tor(G)NL/NL, i.e., torsion lifts.

In our next example all torsion elements are, up to conjugation, containedin K. We give it only for p = 2.

Proposition 9.2 p-cdl-conj Let G be a pro-2 group with pro-2-presentation

G = 〈a, b, t, s | a2 = b2 = [a, b] = [b, at] = atb(ab)−s〉 and F := (s, t)G. Then,setting K := 〈a, b〉, G enjoys the following properties:

(i) G = F×K.

(ii) G is the pro-2 fundamental group of the graph of groups: 〈a, b〉 〈b〉 //

〈c〉=〈at〉##

〈ab〉s=〈bc〉66〈b, c〉

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(iii) F is free pro-2 on the set {t, ta, tb, s, sa}. More precisely, K canbe realized as a group of automorphisms for the free pro-2 groupF (t, y, z, s, u) such that

tb = y, zb = z, sb = uz−1, ub = sz, ta = tz, ya = yz, za = z−1, sa = u

and so that the holomorph F (t, y, z, s, u)×K is isomorphic to G. Theisomorphism is induced naturally by sending K and {t, s} identicallyto K and {t, s}.Moreover, CG(a) = 〈ty−1〉×K, CG(b) = 〈z〉×K and CG(ab) =〈szu−1〉×K.

(iv) All torsion elements in G are, up to conjugation, contained in K.

However 〈b, at〉 ∼= K is a maximal finite subgroup of G not conjugateto K.

(v) Letting “bar” denote passing to the quotient modulo CF (b) = 〈z〉 =〈ata〉 we find that G is a permutational extension corresponding to

the graph of groups: 〈a, b〉 〈b〉 //

〈a〉=〈at〉##

〈ab〉s=〈ab〉66〈a, b〉

Notably CF (a) = 〈t, y〉 does not lift to CF (a) = 〈ty−1〉.However, torsion lifts, i.e., after passing to the quotient, no newtorsion appears.

(vi) Letting “bar” denote passing to the quotient modulo CF (ab) = 〈szu−1〉we find that G is a permutational extension corresponding to the

graph of groups: 〈a, b〉 〈b〉 //

〈c〉=〈at〉##

〈a〉s=〈c〉66〈a, c〉

Notably CF (b) = 〈z, s〉 does not lift to CF (b) = 〈z〉.However, torsion lifts, i.e., after passing to the quotient, no newtorsion appears.

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Proof: (i) is evident and (ii) just reflects the relations in the presentation.(iii) is done by explicit computation – the relation sb = sat−at is helpful. Forchecking (iv) it suffices to see that at and bat are conjugates of elements inK. But this is true for at anyway and it holds for bat = atb = (ab)s. Supposethat 〈a, b〉f = 〈b, at〉. Then we must have af = at and bf = b. Thereforef ∈ 〈z〉. On the other hand CF (a) = 〈ty−1〉 and so f ∈ 〈z〉 ∩ 〈t, y〉 = {1}.So f = 1 and hence 〈b, at〉 = K, a contradiction. Finally, (v) and (vi) resultfrom the induced relations on the respective quotient groups.

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10 1.2.10: Pavel’s suggestion for proving main

result

Theorem 10.1 t-models Let G = F×K be a semidirect product, K afinite p-group, F free pro-p of finite rank, and assume that every torsionelement in G is conjugate to an element in K. Then G is a permutationalextension.

Proof: Suppose that the theorem is false and that G is a counter-examplewith rank(F )+ |K| minimal. It is easy to see that Z(G) = 1. Pick U normalof index p in K, set H := UF .

Claim: One can assume that CF (t) = {1} for every t ∈ U .

By the minimality assumption we have a permutational extension H =HNN (U,Ai, Zi, i ∈ I) q F (Z) for suitable finite sets I and Z and Ai 6= {1}for all i ∈ I. Fix an element k ∈ K \H. Conjugation with k permutes thefinite subgroups of H, and, therefore, for every Ai there is some ik such thatAik is conjugate to Aki in H. Therefore there is an induced action of 〈k〉 uponI.

Let us observe that I decomposes as a disjoint union

I = I ∪ I1 ∪⋃p−1j=0 I

kj

p ,

I := {i ∈ I | CK(Zi) > Ai},I1 := {i ∈ I \ I | ∃h ∈ H,Aki = Ahi },

and Ip is a representative set of the set {i ∈ I |6 ∃h ∈ H,Aki = Ahi } modulothe action of K/U induced by conjugation. We stipulate that for all i ∈ Ipand for j = 0, . . . , p−1 the conjugate Ak

j

i is a representative of its conjugacyclass in H.

Subclaim 1Let P = Φ(K)F , then (i) or (ii) of Lemma 7.6 holds for P .

Proof of Subclaim. During the proof of this claim we write “bar” forpassing to the quotient modulo NP , and we assume that a torsion elementz ∈ G/NH does not satisfy (i) of lemma 7.6, i.e. there exists T ≤ K such thatz ∈ CF (T ) is not conjugate to K by an element of CF (T ). By Lemma 7.6 wecan assume that for T ∈ socle(K), CF (T ) is trivial. Otherwise CK(T ) 6= Kand so there exists a maximal

47

subgroup Γ of G containing 〈P,CK(T )〉.By the minimality assumption on rank(F ) + |K| and since |Γ∩K| < |K|,

one deduces that z is a conjugate in CG(T ) of an element in K.In the same vein we can conclude that whenever z = x for some torsion

element x ∈ G, then CF (x) = CF (z). Hence such z cannot exist in Γ, acontradiction. Thus the subclaim holds.

Subclaim 2For G := G/〈CF (T ) | Cp ∼= T ≤ H〉 every torsion element t of order p is

conjugate to K and CF (t) lifts to CF (t).

Proof of subclaim. If t has order greater then p, then we may assumethat tp

2 ∈ U and so if t 6∈ K denoting by t′ its image in K module F we have〈t, t′〉 centralizes tp

2implying CF (tp

2) 6= 1 a contradiction. So we just need

to prove the result for all t of order p.By the previous subclaim we may assume that CF (t) = 1 for all t ∈ Φ(K).

Fix an element k ∈ K \H of order p. Note that⋃i∈I Zi is k-invariant subset

of the basis of F and so generate a k-invariant free factor L = 〈Zi | i ∈ I〉 ofF on which k acts permutationally. Thus by KST CL(k) is a free factor ofCF (k). It follows that in

M =⊕

i∈I,Ai 6=1

IFp[K/Ai × Zi]⊕MH ,

where MH is a free H-module,⊕

i∈I,Ai 6=1 IFp[K/Ai × Zi] is k-permutationaland we have a direct sum decomposition M1k = M1kL⊕M1k0 where M1kL is asubmodule of

⊕i∈I,Ai 6=1 IFp[K/Ai×Zi] andM1k0∩

⊕i∈I,Ai 6=1 IFp[K/Ai×Zi] = 0.

Thus M/⊕i∈I,Ai 6=1 IFp[K/Ai×Zi] is k-permutational for every such k. Since

M/⊕i∈I,Ai 6=1 IFp[K/Ai×Zi] is exactly the module M obtained from the image

F of F in G := G/〈CF (T ) | Cp ∼= T ≤ H〉 by the factoring out [F , F ].Then F 〈k〉 has only one conjugacy class of groups of order p. Note thatthis also shows that CF (k) maps onto CF (k) since the rank of CF (k) isthe rank of M1k and CF is the rank of M1k0 by Heller-Reiner and the factthat the natural epimorphism F −→ F gives the factorization of M modulo⊕i∈I,Ai 6=1 IFp[K/Ai × Zi]. The subclaim is proved.

We deduce that G := G/〈CF (T ) | Cp ∼= T ≤ H〉 is a permutational exten-

sion, by Theorem ?? modelsstep1 . We want to match the decompositions

of H and G and, to this end, we alter Z. When j ∈ I, then Lemma 4.1l-ivstrengthen-new (ii) shows that we have an epimorphism from CF (Kj)

48

onto CF (Kj). Therefore we can define a retract ϕj : CF (Kj) −→ CF (Kj)and set Yj = ϕj(Y j). Finally, take a retract of F (Y0) inside F and selecta basis Y in there with Y = Y0. We observe that F (Z) ∼= H/(U)H can beconsidered as a subgroup of index p in HNN (K/U,KiU/U, Y i, i ∈ I)qF (Y0)containing Y i, Y0 and so by the Kurosh subgroup theorem (see Theorem 9.1.9

in [13] RZ1 2000 ) can be written as F (Z) ∼=∐j∈I F (Yj) q

∐p−1l=0 F (Y )k

l. So

it allows to replace Z by⋃j∈I Yj ∪

⋃p−1l=0 Y

kl. Then

e-H-perm H = 〈U,Zi | i ∈ I〉 q∐j∈I

F (Yj) qp−1∐l=0

F (Y )kl

. (13)

For proving the Theorem it suffices to construct a permutational extensionG := HNN (K,Pj, Xj, j ∈ J) q F (X) and an isomorphism λ : G → G, thusobtaining a contradiction.

Let J := I ∪ I1 ∪ Ip ∪ I be a disjoint union. For j ∈ I, make Pj :=CK(CF (Aj)) and Xj := Zj; for j ∈ I1 ∪ Ip set Pj := Aj and Xj := Zj as well.If j ∈ I then set Pj := Kj and Xj := Yj. Finally let X := Y and with thesedata form G.

We define λ on⋃j∈J(Pj ∪ Xj) ∪ X ⊆ G to be the map that sends each

Pj and Xj in G identically to the same subgroup, respectively subset in G,and X onto Y . By means of the universal property of the permutationalextension G one can extend λ to become an epimorphism G −→ G.

Rewrite G as a permutational extension G = HNN (B,Ki, Xi, i ∈ I) qF (X), with base groupB := 〈K,Xi | i ∈ I∪I1∪Ip〉. Let B := HNN (B,Ki, Xi, i ∈I). Since H := λ−1(H) is open in G , one can apply the pro-p version of the

Kurosh subgroup theorem (see Theorem 9.1.9 in [13] RZ1 2000 ), taking intoaccount that H contains the normal closure of F (X), to get a decomposition

H = (B ∩H)qp−1∐l=0

F (X)kl

.

Since H ∩Ki = {1} holds for each i ∈ I, Lemma 10 in [7] HZ 07 shows thatH ∩ B can be presented as a free pro-p product of conjugates of H ∩B and afree factor F0. Since all torsion elements in G are conjugate to elements in K,all torsion elements in H are conjugate to elements in U = K∩ H. Thereforeonly a single conjugate of H ∩ B can appear in this free decomposition ofH ∩ B, i.e., H ∩ B = (H ∩B)qF0. It is not difficult to see that (B)B ∩ H =

49

(H ∩ B)H∩B, so F0 = B/(B)B = F (⋃j∈I Xj); therefore we can choose F0 to

have a basis⋃j∈I Xj, so

H = 〈U,Xi | i ∈ I〉 q∐j∈I

F (Xj) qp−1∐l=0

F (X)kl

.

Comparison with Eq.(13 e-H-perm ) implies that the restriction of λ to H

is an isomorphism onto H. Since Z(G) = {1}, the epimorphism λ : G → Gis an isomorphism, a contradiction. The claim holds.

Now the result follows from Theorem 12.6 t-modelsstep1 .

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11 Changes

A major change consists in eliminating Theorem 12.6 t-modelsstep1 and

observing that the original proof of Theorem 10.1 t-models can be adapted:

namely, using Lemma 4.1 l-ivstrengthen-new the induction argument in

Theorem 10.1 t-models works as soon one has a subgroup of index p in Kand in it an element t with CF (t) 6= 1. If such U does not exist then K actsfreely on F and then the claimed result follows already from Theorem 1 in[6] paper3 .

We have decided to include a reworked version of Theorem 12.6 t-modelsstep1

in Sect. 12 s-eliminated (which is is not part of the paper anymore), to-gether with those tools that only are needed for its proof.

We have followed the advices of the report with only one exception,namely 4.11.

References are now given w.r.t. the items of the partial refereeing report.

1.1. We decided to provide reference to Stalling’s 1970 paper, since, in ourpaper, all groups are (topologically) finitely generated.

2.5.– 2.8. Corollary 2.4 c-sub has been reworked. We use M := M/L and

M := M/〈x〉. Also the interpretation of the exact sequence in Eq.(1

e-cohom ) has to be homological.

2.9. concerns Remark 12.2 rem-MN which does not seem to be neededanymore.

2.10. The missing explanation (see Notation 12.3 n-xZ ) concerns the ap-plication of Weiss’ theorem.

2.11. We left the statement in Lemma 12.4 l-modules and simplified itsproof.

2.12.–2.14 Lemma 2.6 l-modules-gen has been corrected and adapted.

3.2. has led to a simplification of our original proof.

3.3.–3.8. we have taken into account.

3.9. Remark 3.5 rem-perm-ext should be clearer now.

51

Comments on 4.1. This is Theorem 12.6 t-modelsstep1 and its proof– the elimated result. The typos and errors should not anymore bepresent. [4.7.] is a helpful argument and we integrated it into theproof. [4.9.] concerns lifting every Zi to a subset of a basis Bi of

CF (Ti) in the proof of Theorem 12.6 t-modelsstep1 . The changesduring the proof address this concern.

4.11. We are thankful for this version of Lemma 4.2. in the report. Wewould have liked to insert it – however, we would have needed that FHbecomes K-invariant. This is not a consequence of F = FH×NH , sinceK can move elements from FH to F \ FH .

4.12.–4.14. have been considered.

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12 What has been eliminated

s-eliminated Later we shall need a result from the Zp-representation theoryof finite p-groups:

Theorem 12.1 t-weiss ([1] weiss A. Weiss) Let G be a p-group, N anormal subgroup of G and M a finitely generated Zp[G]-module. Supposethat M is a free N-module and the submodule MN of fixed elements by N isa permutation lattice for G/N . Then M is a permutation lattice for G.

Remark 12.2 rem-MN Let M be a G-module and N normal in G. Recallthat MN is the largest quotient module of M on which N acts trivially. Notethat MN = M/JM and by Lemma 2.3 l-gp-rg there is an isomorphism of

G-modules from MN onto MN . As a consequence, when M is a finitely gen-erated G-module which is free as an N -module, MN and MN are isomorphicG/N -modules. Therefore, when applying Theorem 12.1 t-weiss we usuallyshall check the hypothesis for MN only.

Notation 12.3 n-xZ For a finite group K let X and Z be finite K-setswith K acting on the left. There is the diagonal action of K on X ×Z givenas k(x, z) := (kx, kz). When the action on X is transitive and on Z istrivial then X × Z, as a K-set, can be identified with K/Kx × Z with Kx

the stabilizer of a given point x ∈ X. We shall consider Zp[K]-permutationmodules Zp[K/A× Z] with A ≤ K and K acting trivially on Z.

Lemma 12.4 l-modules Let M = Zp[K/A × Z] ⊕ N be a direct sumof K-permutational modules, where A ∼= Cp. Then, for A acting by left-multiplication, there is an A-module decomposition M = Mp ⊕M1 with Mp

a free A-module and M1 a trivial A-module containing Zp[CK(A)/A× Z] asa direct summand.

Proof: It suffices to let Z consist of a single element and to assume N ={0}. For every coset rA and a ∈ A we have arA = ara−1A so that leftmultiplication of cosets by elements in A amounts to conjugating them.Fix a left transversal R of K/A. For 1 6= a ∈ A then arA = rA if

and only if r ∈ CK(A). Therefore Zp[K/A] =(⊕

r∈R\CK(A) Zp[rA/A])⊕(⊕

r∈R∩CK(A) Zp[rA/A]). The second summand is a trivial A-module and,

as an A-module, isomorphic to Zp[CK(A)/A]; the first summand is a free

53

A-module: if r 6∈ CK(A) then the coset rA spans over Zp a submodule whichis isomorphic to Zp[A].

We shall use also the following corollary that can be extracted from Corol-lary 7 in [6] paper3 .

Corollary 12.5 c-bases If for each i ∈ I a basis Bi of Hi is given and B is

any basis of H, then⋃i∈I Bi[F, F ]/[F, F ] is a basis of M1 and

⋃p−1j=0 B

cj [F, F ]/[F, F ]

a Zp-basis of Mp. A Zp-basis of Mp−1 is given by {(cic−1)cj | i ∈ I, j =

0, . . . , p− 2}[F, F ]/[F, F ] where ciF = ci0F .

12.1 CF (t) = {1} for torsion elements t in a maximalsubgroup of F×K

Theorem 12.6 t-modelsstep1 Let G = F×K be a semidirect product, withK a finite p-group, F free pro-p of finite rank, and assume that every torsionelement of G is conjugate to an element in K. Let U be a subgroup of index pin K such that CF (t) = {1} for every 1 6= t ∈ U . Then G is a permutationalextension.

Proof: Suppose the Lemma is false. Then there is a counter-example with|K| + rank(F ) minimal. Choose any normal subgroup C of K contained inU ; we shall be particularly interested in C = {1} and C ≤ socle(K).

We claim that G/(C)G satisfies the assumptions of the Theorem. Let usindicate by a bar passing to the quotient modulo (C)G. In case {1} 6= C

Prop.1.7 [24] crelle implies that every torsion element of G lifts to a torsionelement of G and so torsion elements in G are conjugate to torsion elementsin K.

Let H := UF . By Prop. 14 in [6] paper3 the condition CF (t) = 1 for all

t ∈ U implies that H = U q F0 with F0 free pro-p. Therefore H = U q F0.

As a consequence of the Kurosh subgroup theorem (Theorem 9.1.9 [13] RZ1 2000 ),applied to F = (F0)H , we have F =

∐u∈U F

u0 . Hence M := F /[F , F ] is a free

U -module. By Prop.1.7 in [24] crelle every torsion element of G/(U)G liftsto a torsion element of G. The lifted element is conjugate to an element inK and hence the element itself is conjugate to an element in K/U , showingthat there is a single conjugacy class of subgroups of order p in G/(U)G. Let“tilde” denote passing to the quotient modulo [U , F ]. The quotient G/(U)G

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is free pro-p by Cp: from G/(U)G ∼= F×K/U and since F =∐u∈U F

u0 we

can deduce that F ∼= F0 is free pro-p. Then the structure of G/(U)G is de-

scribable as in Lemma 2.6 l-modules-gen (i), i.e., G/(U)G ∼= (Cp× P )q F1,

with P and F1 both free pro-p. Recalling F = F /[U , F ] now part (ii) of thesame lemma implies that MU := F /[F , F ] is K/U -permutational. Hence an

application of Theorem 12.1 t-weiss , taking Remark 12.2 rem-MN into

account, implies that M is K-permutational. This means there exists a fi-nite set I and, for each i ∈ I, a subgroup Ti of K such that M =

⊕i∈I Mi,

where each Mi is, for a suitable positive µi, a µi-fold direct sum of copies ofZp[K/Ti]. Thus M can be rewritten in the form

e-pext-M M =⊕i∈I

Zp[K/Ti × Zi], (14)

where Zi is a set of cardinality µi.Let us prove |Ti| ≤ p for all i ∈ I. We are grateful to the referee for having

suggested the following argument. The decomposition in Eq.(14 e-pext-M )

of M is in particular a decomposition of M considered as a U -module. NowM is a free Zp[U ]-module and Zp[U ] is an indecomposable Zp[U ]-modulesince Zp[U ] is a local ring. By the Krull-Remak-Schmidt theorem each ofthe summands Zp[K/Ti] should be the sum of a number of copies of Zp[U ].Observing that the Zp-rank of Zp[K/Ti] amounts to |K/Ti| whereas the Zp-

rank of Zp[U ] is |U |p

we infer |K : Ti| ≥ |U |, and so |Ti| ≤ p, as claimed.When fixing i ∈ I, then M is a Ti-module. Therefore we can interpret

Eq.(14 e-pext-M ) as a Heller-Reiner decomposition M1 ⊕Mp by putting

M1 := Zp[K/Ti × Zi]Ti and collecting all other summands together with

Zp[Tj × Zj] with j 6= i into Mp. By Corollary 12.5 c-bases there is aTi-Heller-Reiner decomposition M = M ′

1⊕M ′p with M ′

1 the image of CF (Ti).

Since MTi =⊕j∈IM

Tij and, for j 6= i, Mj ≤Mp, we find that

e-uparr MTij ≤ JM ′

jTi , for i 6= j. (15)

Now for M = M ′1⊕M ′

p conclude from Eq.(15 e-uparr ) that M ′1 ≤M1⊕JM .

Then M ′1 + JM = M1 + JM so that the subset Zi + JM of M/JM can be

lifted to become part of a basis Bi freely generating CF (Ti).Let G := HNN (K,Ti, Zi, i ∈ I) with Zi a set of cardinality µi. Let us

show that G ∼= G. There are evident homomorphisms f : K → K ≤ G,

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fi : Ti → Ti ≤ G and bijections gi : Zi → Zi. Since [fi(Ti), gi(zi)] = 1 holdsfor all i ∈ I, and all zi ∈ Zi there is a universal homomorphism ω : G → Gsatisfying ω|K = 1K , ω|Ti

= fi and ω|Zi= gi. Set F := ω−1(F ) and note

that it is a free pro-p normal subgroup of G. By Remark 3.5 rem-perm-ext

M =⊕

i∈I Zp[K/Ti × Zi] and so M and M are isomorphic Zp-modules sincefor all i ∈ I we have |Zi| = |Zi| = µi. As a consequence, the free pro-p groupsF and F have the same rank and so ω|F is an isomorphism. Hence ker (ω) is

finite. Since every finite subgroup of G is conjugate to a subgroup of K wefind ker (ω) ≤ K. But ω|K is the identity map on K showing ker (ω) = {1}.

References

[1] A. Weiss, Rigidity of p-adic p-torsion, Ann. of Math. (2) 127 (1988)317–332.

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