Central automorphisms of finite groups

Abstract

This paper considers an aspect of the general problem of how the structure of a group influences the structure of its automorphisms group. A recent result of Beisiegel shows that if P is a p-group then the central automorphisms group of P has no normal subgroups of order prime to p. So, roughly speaking, most of the central automorphisms are of p-power order. This generalizes an old result of Hopkins that if Aut P is abelian (so every automorphisms is central), then Aut P is a p-group.This paper uses a different approach to consider the case when P is a π-group. It is shown that the central automorphism group of P has a normal. π′-subgroup only if P has an abelian direct factor whose automorphism group has such a subgroup.