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 gl 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 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 (