Outer automorphisms of hypercentral p-groups

Abstract

In his celebrated paper [3] Gaschiitz proved that any finite non-cyclic p-group always admits non-inner automorphisms of order a power of p. In particular this implies that, if G is a finite nilpotent group of order bigger than 2, then Out (G) = Aut(G)/Inn(G) =≠1. Here, as usual, we denote by Aut (G) the full group of automorphisms of G while Inn (G) stands for the group of inner automorphisms, that is automorphisms induced by conjugation by elements of G. After Gaschiitz proved this result, the following question was raised: “if G is an infinite nilpotent group, is it always true that Out (G)≠1?”