On groups admitting a word whose values are Engel

TL;DR

This paper investigates conditions under which certain verbal subgroups in residually finite and locally graded groups are locally nilpotent, focusing on Engel properties of word-values and extending known results.

Contribution

It proves that in residually finite groups with all w-values n-Engel, the verbal subgroup w(G) is locally nilpotent, and extends results to locally graded groups for specific cases.

Findings

Verbal subgroups are locally nilpotent under Engel conditions in residually finite groups.

Confirmed the property for locally graded groups when m=1.

Extended the understanding of Engel conditions in various group classes.

Abstract

Let m, n be positive integers, v a multilinear commutator word and w = v^m. 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.