nLab
holomorph

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

Each group GG embeds into the symmetric group Sym(G)Sym(G) on the underlying set of GG by the left regular representation gl gg\mapsto l_g where l g(h)=ghl_g(h) = g h. The image is isomorphic to GG (that is, the left regular representation of a discrete group is faithful). The normalizer of the image of GG in Sym(G)Sym(G) is called the holomorph.

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

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