ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

On finite groups in which commutators are covered by Engel subgroups

Let {m,n} be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups {G_{1},\ldots,G_{m}} whose union contains all w-values in G. Assume further that all elements of the subgroups {G_{1},\ldots,G_{m}} are n-Engel in G. It is shown that the verbal subgroup {w(G)} is s-Engel for some {\{m,n,w\}} -bounded number s.

Profinite groups with restricted centralizers of commutators

A group G has restricted centralizers if for each g in G the centralizer $C_G(g)$ either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present paper we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.

BFC-theorems for higher commutator subgroups

A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954, B. H. Neumann discovered that if G is a BFC-group then the derived group G′ is finite. Let w=w(x1,…,xn) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if |xG|≤m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If |xw(G)|≤m for every w-value x, then [w(w(G)),w(G)] is finite of order bounded by a function of m and n.

On locally graded groups with a word whose values are Engel

Let m, n be positive integers, let υ be a multilinear commutator word and let w = υm. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

Nonsoluble length of finite groups with commutators of small order

Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length λp(G) is defined as the minimal number of non-p-soluble quotients in a series of this kind.We deal with the question whether, for a given prime p and a given proper group variety , there is a bound for the non-p-soluble length λp of finite groups whose Sylow p-subgroups belong to . Let the word w be a multilinear commutator. In this paper we answer the question in the affirmative in the case where p is odd and the variety is the one of groups satisfying the law we ≡ 1.

WORDS AND PRONILPOTENT SUBGROUPS IN PROFINITE GROUPS

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.