Groupes satisfaisant une condition d'Engel

Let n be a positive integer. We say that a group G satisfies the condition E(n), if every set of n+1 elements of G contains a pair {x,y} such that [x,ky]=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 |GZ*(G)|⩽n113n+2, where Z*(G) is the hypercentre of G.