Amalgamated Products of Profinite Groups: Counterexamples (original) (raw)

On the free profinite products of profinite groups with commuting subgroups

International Journal of Group Theory, 2016

In this paper we introduce the construction of free profinite products of profinite groups with commuting subgroups‎. ‎We study a particular case‎: ‎the proper free profinite products of profinite groups with commuting subgroups‎. ‎We prove some conditions for a free profinite product of profinite groups with commuting subgroups to be proper‎. ‎We derive some consequences‎. ‎We also compute profinite completions of free products of (abstract) groups with commuting subgroups‎.

Algebraic properties of profinite groups

2011

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds which give rise to algebraic properties of topological groups.

On the structure of just infinite profinite groups

Journal of Algebra, 2010

A profinite group G is just infinite if every closed normal subgroup of G is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup H of G, there are only finitely many open normal subgroups of G not contained in H. This extends a result recently established by Barnea, Gavioli, Jaikin-Zapirain, Monti and Scoppola in [1], who proved the same characterisation in the case of pro-p groups. We also use this result to establish a number of features of the general structure of profinite groups with regard to the just infinite property.

Abstract commensurators of profinite groups

Transactions of the American Mathematical Society, 2011

In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group G is a group Comm(G) which depends only on the commensurability class of G. We study various properties of Comm(G); in particular, we find two natural ways to turn it into a topological group. We also use Comm(G) to study topological groups which contain G as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, like Pink's analogue of Mostow's strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups. W ∈ C Cent G (W). By Baire's category theorem, there exists V ∈ C such that Cent G (V) is open in VZ(G) and thus has finite index in VZ(G). Since Cent G (U) ⊇ Cent G (V) whenever U ⊆ V , we conclude that VZ(G) = Cent G (U) for some open subgroup U of G.

Introduction to Profinite Groups

2012

A profinite space / group is the projective limit of finite sets / groups. Galois theory offers a natural frame in order to describe Galois groups as profinite groups. Profinite groups have properties that correspond to some of finite groups: e.g., each profinite group does have p-Sylow subgroups for any prime p. In the same vein, every pro-solvable group (the projective limit of an inverse system of finite solvable groups) has Hall subgroups for any given set of primes. Any group can be equipped with the profinite topology turning it into a topological group. A basis of neighbourhoods of the identity-element consists of all normal subgroups of finite index. Any such group allows a completion w.r.t. this topology – the profinite completion. A free profinite group is the profinite completion of a free group. This can be considered an instance of the amalgamated free product and of the HNN extension (Higman-NeumannNeumann). I do not include cohomological topics in this note.

Elementary equivalence of profinite groups

Bulletin of the London Mathematical Society, 2008

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the classification of the finite simple groups. Our result does not hold any more if the profinite groups are not finitely generated. We give concrete examples of non-isomorphic profinite groups which are elementarily equivalent.

G R ] 2 0 Fe b 20 06 Cartesian products as profinite completions

2008

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.

Profinite Topologies in Free Products of Groups

International Journal of Algebra and Computation, 2004

Let [Formula: see text] be a nonempty class of finite groups closed under taking subgroups, quotients and extensions. We consider groups G endowed with their pro-[Formula: see text] topology, and say that G is 2-subgroup separable if whenever H and K are finitely generated closed subgroups of G, then the subset HK is closed. We prove that if the groups G1 and G2 are 2-subgroup separable, then so is their free product G1*G2. This extends a result to T. Coulbois. The proof uses actions of groups on abstract and profinite trees.

On groups whose subgroups are closed in the profinite topology

Journal of Pure and Applied Algebra, 2009

A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.