On (𝑛 + ½)-Engel groups

Abstract

AbstractLet n be a positive integer. We say that a group G is an {(n+\frac{1}{2})}-Engel group if it satisfies the law {[x,{}_{n}y,x]=1}. The variety of {(n+\frac{1}{2})}-Engel groups lies between the varieties of n-Engel groups and {(n+1)}-Engel groups. In this paper, we study these groups, and in particular, we prove that all {(4+\frac{1}{2})}-Engel {\{2,3\}}-groups are locally nilpotent. We also show that if G is a {(4+\frac{1}{2})}-Engel p-group, where {p\geq 5} is a prime, then {G^{p}} is locally nilpotent.