COMMUTING AUTOMORPHISMS OF AUTO-ENGEL AND AUTO-BELL GROUPS

Let x be an element of a group G, ® 2 Aut(G), n 2 N and A(G) = f® 2 Aut(G)j x®(x) = ®(x)x; 8x 2 Gg. The autocommutator [x;n ®] is defined inductively by [x; ®] = x¡1x® = x¡1®(x) and [x;n+1 ®] = [[x;n ®]; ®]. The group G is said to be n-auto-Engel, if [x;n ®] = [®;n x] = 1, for all x 2 G and every ® 2 Aut(G), where [®; x] = [x; ®]¡1. Also, the group G is called an n-auto-Bell group when [xn; ®] = [x; ®n], for every x 2 G and every ® 2 Aut(G). In this paper, we investigate the properties of such groups and their commuting automorphisms set A(G).