Automorphism groups of some finite groups (Q1326991)

A class of finite groups is shown to have the property that the groups in this class are never full automorphism groups of any finite group. More precisely, there is no finite group \(G\) such that \(| \text{Aut}(G)|\) is an odd number and \(\omega(| \text{Aut}(G)|) \leq 4\). Here if \(n\) is a natural number, \(\omega(n)\) denotes the number of prime divisors of \(n\), counted with multiplicities. As a direct consequence of this result, the following sufficient condition for a finite group \(G\) to have an outer automorphism is obtained: If \(G\) is a group of odd order and if \(\omega(| G/Z(G)|) \leq 4\), then \(G\) has an outer automorphism. The proof uses quite a battery of standard results in group theory.