On Residually Finite Groups with Engel-like Conditions

Abstract

Let m, n be positive integers. Suppose that G is a residually finite group in which for every element x ∈ G there exists a positive integer q = q(x) m such that x q is n-Engel. We show that G is locally virtually nilpotent. Further, let w be a multilinear commutator and G a residually finite group in which for every product of at most 896 w-values x there exists a positive integer q = q(x) dividing m such that x q is n-Engel. Then w(G) is locally virtually nilpotent.2010 Mathematics Subject Classification. 20F45; 20E26.

Bounded Engel elements in residually finite groups

“…An interesting result in this context, due to Wilson [21], states that every n-Engel residually finite group is locally nilpotent. Another result that was deduced following the positive solution of the RBP is that given positive integers m, n, if G is a residually finite group in which for every x ∈ G there exists a positive integer q = q(x) m such that x q is n-Engel, then G is locally virtually nilpotent [1]. We recall that a group possesses a certain property virtually if it has a subgroup of finite index with that property.…”

“…We recall that a group possesses a certain property virtually if it has a subgroup of finite index with that property. For more details concerning Engel elements in residually finite groups see [1,2,3,17,18]. One of the goals of the present article is to study residually finite groups in which some powers are bounded Engel elements.…”