Characterizing inner automorphisms and realizing outer automorphisms

TL;DR

This paper provides elementary, group-theoretical proofs for characterizations of inner automorphisms and constructs finite groups with prescribed outer automorphism groups, advancing understanding of automorphism structures.

Contribution

It offers new elementary proofs of key theorems on automorphisms of finite groups, avoiding graph-theoretical and Lie-theoretical methods.

Findings

Automorphisms are inner iff they extend to all containing groups.

Existence of finite groups with outer automorphism groups isomorphic to any given finite group.

Abstract

We give elementary proofs of the following two theorems on automorphisms of a finite group G: (1) An automorphism of G is inner if and only if it extends to an automorphism of every finite group containing G. (2) There exists a finite group, whose outer automorphism group is isomorphic to G. The first theorem was proved by Pettet using a graph-theoretical construction of Heineken-Liebeck. A Lie-theoretical proof of the second theorem was sketched by Cornulier in a MathOverflow post. Our proofs are purely group-theoretical.