# Outside in

Given a group $G$, embed $G$ in a larger group $H$ such that any automorphism of $G$ is the restriction of an inner automorphism of $H$.

1. Consider the semidirect product $G\rtimes\mathrm{Aut}(G)$, which is called the holomorph of $G$ and denoted $\mathrm{Hol}(G)$. Obviously we have a copy of $G$ in $\mathrm{Hol}(G)$ and the product in $\mathrm{Hol}(G)$ is such that $\varphi g \varphi^{-1} = \varphi(g)$; in other words, in the holomorph, conjugation of a group element by an automorphism amounts to evaluation.