Groups satisfying an Engel condition[Groupes satisfaisant une condition d'Engel]

Abstract

Let n be a positive integer. We say that a group G satisfies the condition ε(n), if every set of n+1 elements of G contains a pair {x, y} such that [x,k y] = 1, for some positive integer k. In this paper, we study finite groups G satisfying this condition. In particular, if G is a finitely generated soluble group, then G/Z*(G) ≤n113√n+2, where Z*(G) is the hypercentre of G. © 2004 Elsevier Inc. All rights reserved.