holomorph

One would like to embed an abstract group into a bigger group $K$ in which every automorphism of $G$ is obtained by restricting (to $G$) an inner automorphism of $K$ that fixes $G$ as a subset of $K$. The holomorph is the universal (smallest) solution to this problem.

Each group $G$ embeds into the symmetric group $\mathrm{Sym}(G)$ on the underlying set of $G$ by the left regular representation $g\mapsto {l}_{g}$ where ${l}_{g}(h)=gh$. The image is isomorphic to $G$ (that is, the left regular representation of a discrete group is faithful). The normalizer of the image of $G$ in $\mathrm{Sym}(G)$ is called the **holomorph**.

The holomorph occurs very naturally as the group of arrows of the 2-group (groupoid internal to $\mathrm{Groups}$).

Revised on February 4, 2010 15:21:25
by Tim Porter
(95.147.237.50)