On varieties of groups satisfying an Engel type identity

TL;DR

This paper investigates varieties of groups defined by Engel conditions and properties of verbal subgroups, establishing new classifications and conditions under which these classes form varieties.

Contribution

It proves new theorems characterizing when certain groups with Engel and verbal subgroup conditions form varieties, including cases with prime-power and arbitrary integers.

Findings

The class of groups with n-Engel w-values and locally nilpotent w(G) is a variety.

For prime-power m, groups with n-Engel w-values and a specific normal series form a variety.

Extends results to arbitrary m with complex verbal subgroup conditions, establishing new variety classes.

Abstract

Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result remains valid with m allowed to be an arbitrary positive integer. In this direction, we show that if m, n are positive integers, u a multilinear commutator word and v the product of 896 u-words, then the class of all groups G in which the v^m-values are n-Engel and the verbal subgroup u(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem C).