Groups generated by a finite Engel set

A subset $S$ of a group $G$ is called an Engel set if, for all $x,y\in S$, there is a non-negative integer $n=n(x,y)$ such that [x,\,_n y]=1. In this paper we are interested in finding conditions for a group generated by a finite Engel set to be nilpotent. In particular, we focus our investigation on groups generated by an Engel set of size two.Comment: to appear in Journal of Algebr

Monatshefte fur Mathematik

Texto completo: acesso restrito. p. 151-159For a group G , denote by ω(G) the number of conjugacy classes of normalizers of subgroups of G . Clearly, ω(G)=1 if and only if G is a Dedekind group. Hence if G is a 2-group, then G is nilpotent of class ≤2 and if G is a p -group, p>2 , then G is abelian. We prove a generalization of this. Let G be a finite p -group with ω(G)≤p+1 . If p=2 , then G is of class ≤3 ; if p>2 , then G is of class ≤2