SOME PROPERTIES OF 2-AUTO-ENGEL GROUPS

摘要

For a given group G, with an element x in G and automorphism α in Aut(G), the nth autocommutator [x,n a] is defined recursively by [x,a] = x-1xa and [x,n α] = [[x,n-1 α], α] for all n > 1. The group G is said to be n-auto-Engel if [x,n α] = [a,n x] = 1, for all x ∈ G and all a ∈ Aut(G), where [a,x] = [x, α]-1. We study the structure of 2-auto-Engel groups and show that 2-auto-Engel groups indeed satisfy the equation a(x)a-1(x) = x2, for all x ∈ G and a ∈ Aut(G). Also, we give a precise description of all abelian 2-auto-Engel groups of finite 2-rank as well as 2-auto-Engel 2-groups, which are not purely non-abelian, and construct an infinite family of purely non-abelian 2-auto-Engel 2-groups.