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.

A NILPOTENCY CRITERION FOR SOME VERBAL SUBGROUPS

The word $w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$ is a simple commutator word if $k\geq 2,i_{1}\neq i_{2}$ and $i_{j}\in \{1,\ldots ,m\}$ for some $m>1$. For a finite group $G$, we prove that if $i_{1}\neq i_{j}$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite.

On the width of verbal subgroups of the groups of triangular matrices over a field of arbitrary characteristic

The width [Formula: see text] of the verbal subgroup [Formula: see text] of a group [Formula: see text] defined by a collection of group words [Formula: see text] is the smallest number [Formula: see text] in [Formula: see text] such that every element of [Formula: see text] is the product of at most [Formula: see text] words in [Formula: see text] evaluated on [Formula: see text] and their inverses. Well known that every verbal subgroup of the group [Formula: see text] of triangular matrices over an arbitrary field [Formula: see text] can be defined by just one word: an outer commutator word or a power word. We prove that [Formula: see text] for every outer commutator word [Formula: see text] and that [Formula: see text] except for two cases, when it is equal to 2. For finitary triangular groups, the situation is similar.

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.

On the probability distribution associated to commutator word map in finite groups

In this paper, we study the probability distribution associated to the commutator word map. In other words, we study the probability of a given element of a group to be equal to a commutator of two randomly chosen group elements. We compute explicit formulas for calculating this probability for some interesting classes of groups having only two different conjugacy class sizes.

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.

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.