On locally graded groups with a word whose values are Engel

TL;DR

This paper proves that in locally graded groups, if a certain word's values are Engel, then the associated verbal subgroup is locally nilpotent, advancing understanding of Engel conditions in group theory.

Contribution

It establishes that for locally graded groups with all w-values being n-Engel, the verbal subgroup w(G) is necessarily locally nilpotent, a new result linking Engel conditions to local nilpotency.

Findings

w(G) is locally nilpotent under given conditions

All w-values are n-Engel in the group

Verbal subgroup w(G) inherits local nilpotency

Abstract

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