On groups admitting a word whose values are Engel

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.

Cited by in corpus (6)

• The non-abelian tensor square of residually finite groups
• On residually finite groups with Engel-like conditions
• Bounded Engel elements in residually finite groups
• On varieties of groups satisfying an Engel type identity
• On residually finite groups satisfying an Engel type identity
• An Engel condition for orderable groups