We give a definition of weakly sofic groups (w-sofic groups). Our definition is rather natural extension of the definition of sofic groups where instead of Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non w-sofic groups is equivalent to some conjecture about profinite topology on free groups.

We demonstrate that the blocks of a profinite group whose defect groups are cyclic have Brauer tree algebra structure analogous to case finite groups. show further block with Z p $\mathbb {Z}_{p}$ is star type.

Given a finite group G and an element g∈G, we may compare the expected number e(G) of elements needed to generate e(G,g) together with g. We address following question: how large can difference e(G)−e(G,g) be? prove that in general this be arbitrarily large. For example for every positive integer n there exists 2-generated such e(G)≥n but e(G,g)≤5 some g∈G. However, if derived subgroup is nilpo...

A group word w is said to be strongly concise in a class C $\mathcal C$ of profinite groups if, for any G , either takes at least continuum many values or the verbal subgroup ( ) $w(G)$ finite. It conjectured that all words are groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture multilinear commutator words, as well some other particular words. They also every nilpotent group...

One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an...

THEOREM. Let F be the free profinite group on a set X, where \X\ > 2, and let n be a non-empty set of primes. Then F has a maximal abelian subgroup isomorphic to HpEn Zp. The idea of the proof is the following: we show that A — Ylpe7I1p is a free factor of Pa, i.e. fia ^ A *B for some profinite group B. To conclude from this that A is a maximal abelian subgroup of Fa (the general case then foll...

The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds placed on its complexity as expressed by its scattered height.