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Subgroups of free pro pproducts

Wolfgang Herfort and Luis Ribes

Mathematical Proceedings of the Cambridge Philosophical Society / Volume 101 / Issue 02 / March 1987, pp 197  206DOI: 10.1017/S0305004100066548, Published online: 24 October 2008

Link to this article: http://journals.cambridge.org/abstract_S0305004100066548

How to cite this article:Wolfgang Herfort and Luis Ribes (1987). Subgroups of free pro pproducts. Mathematical Proceedings of the Cambridge Philosophical Society, 101, pp 197206 doi:10.1017/S0305004100066548

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Math. Proc. Camb. Phil. Soc. (1987), 101, 197 197Printed in Great Britain

Subgroups of free pro -p-products

B Y WOLFGANG HERFORT

Institutfilr Numerische und Angewandte Mathematik,Technische Universitdt Wien, Austria

AND LUIS RIBES

Department of Mathematics and Statistics, Carleton University,Ottawa, Ontario KlS 556, Canada

(Received 3 February 1986; revised 9 April 1986)

1. Introduction

HF is a free profinite group, it is well known that the closed subgroups of F need notbe free profinite; however, if p is a prime number, every closed subgroup of a free pro-#-group is free pro-j? (cf. [2, 8, 7]). In this paper we show that there is an analogouscontrast regarding the closed subgroups of free products in the category of profinitegroups, and the closed subgroups of free products in the category of pro-p-groups, atleast for (topologically) finitely generated subgroups.

For a subgroup H of a free product G = 11̂ 4.̂ in some category of groups, the questionis whether one can express H as a free product, in the same category, of groups relatedin some reasonable way to the free factors Ai of G, taking the Kurosh theorem for freeproducts of abstract groups as a model. This is the case, for example, if G is a freeproduct in the category of profinite groups, and H is an open subgroup of G (cf.[1, 3, 4)]; however it is obviously not the case for every closed subgroup H of G(e.g. take H to be a non-trivial pro-p-subgroup of G).

In this paper we prove that if p is a prime number, G = II Ai is a free product in thecategory of pro-p-groups, and H is a (topologically) finitely generated closed subgroupof G, then H is a free product, in the category of pro-p-groups, of a free pro-^-groupand certain subgroups of conjugates of the factors At (see Theorem 4-4 for a precisestatement).

Note that if each Ai is Zp (the additive group of the ring of #-adic integers), then Gis a free pro-#-group, and our statement reduces to a well-known theorem of Tate(cf. [2]) that asserts a closed subgroup of a free pro-^-group is free pro-p.

The proof of the result uses heavily the fact that H is finitely generated, essentiallyin the following form: every (continuous) endomorphism of H is an isomorphism. Inaddition, the main ingredients for the proof are

(i) the known structure of H when it is open in G; and(ii) the location of the finite subgroups of a free pro-^-product of pro-^J-groups ([5],

theorem 2).

The proof proceeds in two steps. First (Proposition 4-1) we establish the structure ofa finitely generated subgroup of a free product of finitely many finite ^-groups. Thenwe extend this result in two directions: on the one hand we allow the factors in thefree product to be infinite pro-^)-groups, and on the other we do not impose any

198 WOLFGANG HERFORT AND LUIS RIBES

restrictions on the number of factors of the free products; they could be indexed by aninfinite set, or even by a topological space in the sense of [3] or [4].

2. Notation and terminology

All groups in this paper are profinite, all subgroups are closed, and all homo-morphisms continuous. Throughout the paper, p represents a fixed prime number.Our results are concerned mainly with pro-^-groups; however, whenever possible wehave stated some auxiliary results under less restrictive hypotheses. By # we alwaysmean a class of finite groups closed under subgroups, quotients and extensions. A pro-*

Subgroups of free pro-p-products 199

epimorphism HN/(HN)*^-H/H* given by hn(HN)*\->hH*. This map splits bymea,mo{thema,vp:H/H*->HN/(HN)*givenbyhH*y-+h(HN)*(sinceH* < (HN)*).It follows that p is 1-1, so a basis of H/H* can be extended to a basis of HN/(HN)*,and also H n (HN)* = H*. \

LEMMA 3-2. Let G = IIJLi-^i be a free pro-^-product of pro-'tf-groups At. Let s,teGbe such that A\ n A\ =j= 1. Then i = j , and A\ = A).

Proof. Let K be the cartesian subgroup of G (cf. [5], §2). Note first that A\ < AtK.So if 1 =(= xeA\ n A\, then a; = a ^ = a;-fc2> 1 4 = ^ 6 ^ , 1 4=OyG^, and kx,h2eK.So â = â , and in particular i = j . Now, since A\ n -4J 4= 1, it follows from Theorem B'in [5] that ts'1 e At. Thus A% = A). \

LEMMA 3-3. Let G = (I l i^i) HP be a free pro-t?-product of pro-'S-groups, with eachAt finite and F torsion-free. Let H ^ G and let A be a maximal finite subgroup ofH. Thenthere exists an open normal subgroup No in G such that A is a maximal finite subgroup ofHN0. Moreover, suppose , ....Aft

1) = Ui=i^?*- In particular, each A\* is afree factor ofG.

Proof. Put Gx = (A{\ ...,AB

n"). Then GXG* = G, since a" is congruent to amodG*,for any a in A{ and g in G. Hence G1 = G. Define p: G->G as sending At onto .4?', inthe obvious way. Since G is finitely generated and p is onto, we have that p is an iso-morphism (cf. [7], p. 68); thus G = II Ap. \

COROLLARY 3-5. Let G = II?=i4i be a free pro-p-product, where the A{'s are either finiteor finitely generated torsion-free pro-p-groups. If A is a maximal finite subgroup of G,then A is conjugate to one of the At, and in fact A is a free factor of G.

Proof. Follows from theorem 2 in [5], and Lemma 3-4. |

LEMMA 3-6. Let G = ]Jf=iAi be a free pro-p-product of finite and finitely generatedtorsion-free pro-p-groups, and let H < G. If A is a maximal finite subgroup of H, thenthere exists an open normal subgroup No of G such that, for each open normal subgroup Nof G with N < N0,Aisa free factor of NH.

Proof. This follows from the above corollary, Theorem K, and Lemma 3-3. |

200 WOLFGANG H E R F O B T AND L U I S R I B E S

LEMMA 3-7. Let G = W2=i-^i oe a free pro-p-product of finite and finitely generatedtorsion-free pro-p-groups, and let H < 6r, Assume Mv ...,Mr are maximal finite subgroupsof H such that Mt n Mf = I, for all different i, j . Let T = (Mj Mr). Then

T = IK-i-M;.

Moreover, there exists an open normal subgroup No of G such that for every open normalsubgroup N < No ofG, T is a free factor of HN.

Proof. Since Mt — {1} and Mf are compact and disjoint, there exists an open normalsubgroup No of G such that Mi — {1} and Mf

N° are disjoint for distinct i and j , i.e.Mt n Mf

N" = 1. Using Lemma 3-3 we choose No so that, in addition, Mt is a maximalfinite subgroup o£HN0, for each i. Let N < No be an open normal subgroup of G; thenby Theorem K,

Then there exist tt in HN such that M.i = (A^W n HN)*i for suitable u(i). Note thatfor different i's the factors Afyffl n HN are distinct since Mt n i f f ^ = 1. The resultfollows then from Lemma 3-4. |

LEMMA 3-8. Let G = AUB be a free pro-p-product of finitely generated pro-p-groups.Then

B* = BnG*.

Proof (due to J.Wilson). Since for a pro-^p-group K one has K* = KV[K,K], itfollows that the kernel of the obvious map G = A II B^-B^-B/B* contains G*. Onthe other hand, each element of B - B* has non-trivial image, and so cannot be inG*. Hence B f]G* ^ B*. The other inclusion is obvious. |

Remarks. (1) We also have B* = Bf\G* HA and B are pro-p-groups and G = A x B.(2) In general, if G, H are pro-p-groups and H < G, then a minimal set of generators

of H can be extended to a minimal set of generators of G if and only if H n G* = H*.

LEMMA 3-9. Let G = (LT™=1 î) II -R 6e a free pro-p-product of finite p-groups Ait anda finitely generated torsion-free pro-p-group R, and let H be a finitely generated subgroupof G. Let ^ be a maximal set of maximal finite subgroups of H such that if M and S aredifferent groups in

Subgroups of free pro--p-products 201

in [5] and Lemma 3-2, for some 5 in HN one has #? = D. Then by Lemma 3-2 againM\ = Ml and therefore D = MJ; so L is in M\. \

LEMMA 3-10. Let G = Ut=1Ai be a free pro-p-product of finite p-groups At. Let H bea finitely generated subgroup of G. Then H = TUF, where T is as in Lemma 3-9 and Fis a free pro-p-group. Moreover, there exists an open normal subgroup V of G such that,for each open normal subgroup S of G with S < V, H is a free factor of HS, and eachmaximal finite subgroup of H is a maximal finite subgroup of HS.

Proof. By Lemma 3-9, T = Jl{Mi where ^ = {Mu...,Mt] is a maximal set ofmaximal finite subgroups of H such that Mf n Mt = 1, for i 4=j. By Lemmas 3-1 and3-7, there exists an open normal subgroup N of G such that T is a free factor of HN,and (HN)* (\H = H*. Now, since T* = (HN)* n T (cf. Lemma 3-8), it follows thatT* = H* n T, and so T/T* is embedded in H/H*. Let X £ T be a minimal set ofgenerators of T. Then there is a subset Y of H such that X fcl F is a minimal set ofgenerators of H. Define F to be the subgroup of H generated by Y.

By Theorem K and Lemma 3-3, HN = T 11(11^) UFN, where each Bt is finite, andFN is a free pro-#-group. Clearly, by theorem 2 in [5] and Lemma 3-9 (iii), we haveH n BfN = 1, for each i. Therefore, there exists an open normal subgroup V of G,V s? N, such that HV n 2 ? ^ = 1, for each i. Let S < F be an open normal subgroupof G. Then (again using Theorem K applied to HN)

HS=(U Mt) u (uMpnHN) UF

where Fs is a free pro-p-group and atieHN.Now, let Z = {zv ...,zr} c {j^Mp

!(\HN be a minimal set of generators forUijMiV n #iV. Then we claim that the set X U Y U Z is linarly independent modulo(HS)*. For otherwise, there exist a.i,f}i,yie2,/p7j, not all zero, with

a*1 - K ^ i 1 -yfczV-*?'e(##)* ^ (HN)*.

But zi e T(HN)*, for each t. So

for some ^ e Z/^Z. However, z1,...,xn,y1,..., ym are linearly independent modulo H*hence & = ... = J3m = 0. Thus a?

1... ^ z { ' . . . zyrre(HS)*, and so

since X u Y is linearly independent modulo (HS)*. This proves the claim. Nowextend X u Y U Z to a minimal set of generators X u F u £ u ? 7 o f HS, where

By proposition 2-9 in [6], we have | Y\ + | U\ = d(Fs). Let x'lt..., x'n, u[,..., u'd be a basisfor Fs. Define a homomorphism p: HS^-HS by sending T identically to T, Ui-Bfidentically to ]JiBi, x't to x{ (i = 1,...,«) and M̂ to uf (j = 1, ...,d). Since/9 is an ontoendomorphism, it is an isomorphism (cf. [7], p. 68). I t follows that F is a free pro-^j-group and HS = TII (lli-B*) UF U(Uy. Thus H = T MF, and H is a free factor ofHS. |

202 WOLFGANG HERFORT AND LUIS RIBES

4. Subgroups of free products

In this section we prove our main result. We begin with the special case when thefactors of the free product are finite ̂ -groups and there are only finitely many factors.

PROPOSITION 4-1. Let O = II*=i-

Subgroups of free pro-jp-products 203

LEMMA 4-2. Let Ax be pro-%'-groups indexed by a pointed, compact, Hausdorff, totallydisconnected topological space (X,*) (c/. [3, 4]). Let G = llx-^z *e their free pro-%-product. Then

G = lim II AxR(U)XIR

where %isa cofinal set of open normal subgroups ofG, Si is a cofinal set of open and closedequivalence relations on X, U e^t, Re Si, and AxR(U) = AxR[AxR n R(U) where, foreachReSH, B: G->GR is the canonical epimorphism and GR =

Proof. The epimorphisms

II II AxR(U)X>R

p: G-^lim u AxR{U).•XIR

induce an epimorphism

Conversely, given an open normal subgroup V of G, the canonical map ijrv: G^-G/Vfactors through some R: G->GR; hence p factors through G->GR/ U AxR/AxS(\tBV-+GB/tBV+G/V.XIR XIR

These in turn induce a homomorphism

y.Km U AxR(U)^G.XIR

G/V -*- lim II AxR{U)

R~ UAxR(V)

We know p is onto. For each V, one has i/rvwp = rjrv. So yp = idG. Thus p is 1-1,and therefore p is an isomorphism. |

We are now in a good position to prove the main theorem, by combining the resultsof Proposition 4-1 and Lemma 4-2. First, we state and prove a possibly more generalresult than we really need.

PROPOSITION 4-3. Let G = lim U Ainbe a protective limit of pro-p-products of finite

p-groups Ain. Assume

(1) Each set Nt is finite, and {N^r}^} is an inverse system of sets.(2) For i>j (i,jel), the canonical epimorphism viS: 11neNiAin^-U.meN.Aim is

induced by epimorphisms Ain^ Ajm, where m = rj^n).For each v = (n(i))eN = limiV ,̂ put AB = lim.4in(f). Let H be a finitely generated

subgroup of G. Then there is a finite subset of M of N such that

204 W O L F G A N G H B B F O B T AND L U I S E I B E S

where, for each, ft, a(/i, k) runs through a complete set of double coset representatives of Aand H in G, and where F is a free pro-p-group. Moreover, if'veN— M,then A * (]H = 1,for all oceG.

Remarks. (1) In the proof this proposition we will make frequent use of the analogueof Grushko's theorem for pro-^-groups: if A = BIIC is a pro-p-product then

d(A) = d{B) + d{C)

(cf. [6], proposition 2-9). This implies, in particular, that in the statement of theproposition, A**-^ n H = 1 for almost all (/i, k).

(2) Also, we will refer often to the Hopfian property of a finitely generated profinitegroup A: every epimorphism f: A->A must be an isomorphism (cf. [7], p. 68).

Proof of the proposition. Denote by pt: G^>\[neNiAin = Gt the canonical epi-morphism, (iel). Set Hi = pi{H). Since ^ = l i m ^ , and d(H) > d(Hi) > d(H})

(i,jel, i >j), we may assume d(H) — d(J5 )̂, for all iel, substituting / by a cofinalsubset if necessary. By Proposition 4-1,

where {a(i,n, k)\k} is a complete set of double coset representatives of Ain and Ht inGit and Ft is a free pro-^-group. Since 7/ij(A^-

n-k) n Ht) is finite, it is conjugate to asubgroup of some Affim-l) n Hp m = %(«) (cf. [5], theorem 2). So,

n,kand Ft

have trivial intersection modulo Hf = Hf [H^, Hj], the Frattini subgroup of H^. I tfollows that d(Fi) ^ d(Fj-); and since d(Ft) is bounded by d(H), we may assume (takinga cofinal subset of/) that d(Fi) is constant for all iel. Therefore we may also assumethat the number of non-trivial free factors in the decomposition (*) is constant for alliel. Hence A$'n-k> n Ht + 1 implies 7}ij(A°$-

n'k'> n Ht) is non-trivial and it is con-jugate in Hj to some A^m'l) n Hjt m = T}i}(n). One then deduces (see Lemma 3-4 fora similar argument) from the Hopfian property of Hp that

n,k

Say d(Fi) = r (constant!). Let Yi be a minimal set of generators for ]lniic-^-in'n'k) n Ht,

and let £^ be the set of r-tuples (x\,..., xl) of elements of Ht such that Yt u {x\,..., xtyis a minimal set of generators for Ht. I t follows from the Hopfian property thatFt « (x{, ...,xi); moreover, (x\,...,x

{r)eS^ implies ( ^ z j , . . . , ^ a ; * ) e ^ . Let

(x1,...,xT)e\imS?i;i

then xlt..., xr eH, and for each ieI,Ft « (piXx,...,piXr). So by the Hopfian propertyof Ht we may assume Ft = (PiXlt ..^p^). Therefore we have %3(i^) = Fp wheneveri > j . Moreover, F : = (xv ..., xr) is a free pro-p subgroup of H of rank r. Turning nowto the other factors of Hit note first that ifr/ii(Ai£-

n'k) n HJ is conjugate to

Subgroups of free pro-ip-products 205

then, in fact, ^i3(a(i, n, k)) and a(j, m, I) belong to the same double coset of Aim and Hpfor if ^(a(»,»,&)) = aa(j,m,t)h, aeAis, heHp then

thus oc(j,m,t) = a(j,m,l) (cf. [5], theorem 2). Label a double coset Aina(i,n,k)Htgood ifAi£-n-k) n Ht #= 1. The above remark shows that rj^ sends good double cosets togood ones, in fact v^ defines a bijective correspondence between the sets of good doublecosets in Ht and Hp since the number of good double cosets is finite and constant forall iel. Let {Ain(i)a(i,n(i), tyH^i} be a compatible set of good double cosets. ThenVnin(i)) = n(j)- Put v = (n(i)) elimNt. Changing the notation, if necessary, we may

assume that the index k is the same for all i. Choose a(v, k) elim (Ain{i)a(i, n(i), k)Ht)i

(note that this last inverse limit is not empty since each double coset is compact andnon-empty). Clearly if A; 4= t, the elements a{v, k) and a{v, t) (if they have been defined)represent different double cosets of Av: = lim Ain(i) and H = lim Ht in G. Using the

Hopfian property of Hit we may assume that pt(a(v, k)) is the original double cosetrepresentative of p^A^p^aiy, k))Ht as used in the decomposition (*). Moreover, fora compatible set {AinWa(i, n(i), k)H^, we know

and soAfv-k) n H = lim (pi(^J,)«

206 WOLFGANG HERFORT AND LUIS RIBES

Proof. This follows immediately from Proposition 4-3 and Lemma 4-2. |Remark. We emphasize that Theorem 4-4 holds in two important special cases.

First, X could be finite, and G the ordinary free pro-_p-product of the possibly infinitepro-jp-groups Ax. Second, the groups Ax could be indexed by any set, and G would bethen their free pro-jp-product, in the sense of [1]; in this last case the space (X, *) inthe theorem is the one-point compactification of the indexing set, thought of as adiscrete topological space (cf. [3]).

Question. The methods used here need very strongly the finite generation hypothesisof H. What is a good description, in the spirit of the Kurosh subgroup theorem forabstract groups, of a subgroup of a free pro-#-product, if the subgroup is not finitelygenerated ?

The results of this paper were announced at the Kertesz-Szele Group TheoryColloquium, Debrecen, September 1985.

Added in proof. D. Haran has recently obtained a description of the countablygenerated subgroups of free pro-p-products using a new concept of free product.(Cf. On closed subgroups of free products of profinite groups, manuscript.)

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