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 semiconcise words

Let w be a group-word. For a group G, let {G_{w}} denote the set of all w-values in G and {w(G)} the verbal subgroup of G corresponding to w. The word w is semiconcise if the subgroup {[w(G),G]} is finite whenever {G_{w}} is finite. The group G is an {\mathrm{FC}(w)}-group if the set of conjugates {x^{G_{w}}} is finite for all {x\in G}. We prove that if w is a semiconcise word and G is an {\mathrm{FC}(w)}-group, then the subgroup {[w(G),G]} is {\mathrm{FC}}-embedded in G, that is, the intersection {C_{G}(x)\cap[w(G),G]} has finite index in {[w(G),G]} for all {x\in G}. A similar result holds for {\mathrm{BFC}(w)}-groups, that are groups in which the sets {x^{G_{w}}} are boundedly finite. We also show that this is no longer true if w is not semiconcise.

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.

Words of Engel type are concise in residually finite groups

Given a group-word [Formula: see text] and a group [Formula: see text], the verbal subgroup [Formula: see text] is the one generated by all [Formula: see text]-values in [Formula: see text]. The word [Formula: see text] is said to be concise if [Formula: see text] is finite whenever the set of [Formula: see text]-values in [Formula: see text] is finite. In 1960s, Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall’s question remains wide open in the class of residually finite groups. In the present paper we show that various generalizations of the Engel word are concise in residually finite groups.

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.

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 GROUPS WITH FINITE CONJUGACY CLASSES IN A VERBAL SUBGROUP

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly 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.