On prosupersolvable groups (original) (raw)

2010

In this paper we prove that if G is a finitely generated pro-(finite nilpotent) group, then every subgroup G" , generated by nth powers of elements of G, is closed in G. It is also obtained, as a consequence of the above proof, that if G is a nilpotent group generated by m elements xt,..., xm, then there is a function f(m, n) such that if every word in x*1 of length < f(m, n) has order n , then G is a group of exponent n . This question had been formulated by Ol'shansky in the general case and, in this paper, is proved in the solvable case and the problem is reduced to the existence of such function for finite simple groups. A group G is called residually finite if it has a family of normal subgroups {Ni}iei such that G/N is a finite group and f]i€I N = 1. The set of normal subgroups may be taken as basis of a topology over G. So a profinite group is a residually finite group that is complete with respect to the above topology, that is, an inverse limit of finite 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.

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.

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.

The profinite completion of accessible groups

Cornell University - arXiv, 2022

We introduce a class A of finitely generated residually finite accessible groups with some natural restriction on one-ended vertex groups in their JSJ-decompositions. We prove that the profinite completion of groups in A almost detects its JSJ-decomposition and compute the genus of free products of groups in A.

On power subgroups of profinite groups

Transactions of the American Mathematical Society, 1994

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.

On profinite groups with finite abelianizations

Selecta Mathematica, 2007

Profinite groups with finite p-abelianizations arise in various contexts: group theory, number theory and geometry. Using Ph. Furtwängler's transfer vanishing theorem it will be proved that a finitely generated profinite groupĜ with this property satisfies H 1 (Ĝ, F p [[Ĝ]]) = 0 (Thm. A). As a consequence one finds that a hereditarily just-infinite non-virtually cyclic prop group has only one end (Cor. B). Applied to 3-dimensional Poincaré duality groups, Theorem A yields a generalization of A. Reznikov's theorem on 3-dimensional co-compact hyperbolic lattices violating W. Thurston's conjecture (Thm. C).

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 prosupersolvable groups (original) (raw)

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