Characterizing inner automorphisms and realizing outer automorphisms

2024/05/05 by Benjamin Sambale, Sambale, Benjamin · 2 citations
Mathematics · #Advanced Topics in Algebra #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2405.02992

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.

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• Global representation theory: Homological foundations
• Subgroups with all finite lifts isomorphic are conjugate

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