A characterization of inner automorphisms

Abstract:

It turns out that one can characterize inner automorphisms without mentioning either conjugation or specific elements. We prove the following Theorem Let $G$ be a group and let $\alpha$ be an automorphism of $G$. The automorphism $\alpha$ is an inner automorphism of $G$ if and only if $\alpha$ has the property that whenever $G$ is embedded in a group $H$, then $\alpha$ extends to some automorphism of $H$.