An improved characterisation of inner automorphisms of groups

TL;DR

This paper demonstrates that every group can be embedded into a special type of group where only inner automorphisms extend to endomorphisms, providing a new characterization of inner automorphisms.

Contribution

It introduces a novel embedding of groups into simple, complete, co-Hopfian groups that characterizes inner automorphisms through endomorphism extension.

Findings

Every group embeds malnormally into a simple, complete, co-Hopfian group.

Non-trivial endomorphisms extend to the larger group if and only if they are inner automorphisms.

Strengthens a theorem of Schupp and answers a question of Bergman.

Abstract

We show that every group $G$ embeds malnormally into a simple, complete co-Hopfian group $H$. This implies that a non-trivial endomorphism of $G$ extends to $H$ if and only if it is an inner automorphism, strengthening a theorem of Schupp and answering a question of Bergman.