Pavel Shumyatsky

ON GROUPS WITH COMMUTATORS OF BOUNDED ORDER

Let p be a prime, k a non-negative integer. We prove that if G is a residually finite group such that [x, y]p k = 1 for all x, y ∈ G, then G′ is locally finite.

Fixed points of Frobenius groups of automorphisms

A finite group admits a Frobenius automorphisms group FH with a kernel and complement H such that the fixed-point subgroup of F is trivial. It is further proved that every FH-invariant elementary Abelian section of G is a free module for an appropriate prime p. The exponent of a group is bounded with a metacyclic Frobenius group of automorphisms and it is supposed that a finite Frobenius group FH with cyclic kernel F and complement H acts on a finite group G. Bounds for the nilpotency class of groups and Lie rings admitting a metacyclic Frobenius group of automorphisms with fixed-point free kernel are obtained. It is also found that a locally nilpotent torsion-free group G admits a finite Frobenius group of automorphisms FH with cyclic kernel F and complement H of order q.

Derived subgroups of fixed points

LetA be an elementary abelianq-group acting on a finiteq′-groupG. We show that ifA has rank at least 3, then properties ofCG(a)′, 1 ≠a ∈A restrict the structure ofG′. In particular, we consider exponent, order, rank and number of generators.

On varieties arising from the solution of the Restricted Burnside Problem

Abstract For a family of group-words w we prove that the class of all groups G satisfying the identity w n ≡1 and having the verbal subgroup w ( G ) locally nilpotent is a variety.

Profinite groups and the fixed points of coprime automorphisms

Abstract The main result of the paper is the following theorem. Let q be a prime and A an elementary abelian group of order q 3 . Suppose that A acts coprimely on a profinite group G and assume that C G ( a ) is locally nilpotent for each a ∈ A # . Then the group G is locally nilpotent.

On groups admitting a word whose values are Engel

Let m, n be positive integers, v a multilinear commutator word and w = vm. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.

A focal subgroup theorem for outer commutator words

Let

Boundedly generated subgroups of finite groups

G

Groups with bounded verbal conjugacy classes

be a finite group of order

A (locally nilpotent)-by-nilpotent variety of groups

p^am