# nLab 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 $Sym(G)$ on the underlying set of $G$ by the left regular representation $g\mapsto l_g$ where $l_g(h) = g h$. 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 $Sym(G)$ is called the holomorph.

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

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