nLab
normalizer

Contents

Definition

Given a subset SS of a group GG, its normalizer N(S)=N G(S)N(S)=N_G(S) is the subgroup of GG consisting of all elements gGg\in G such that gS=Sgg S = S g, i.e. for each sSs\in S there is sSs'\in S such that gs=sgg s=s'g.

If SS is itself a subgroup, then SS is a normal subgroup of N G(S)N_G(S); moreover N G(S)N_G(S) is the largest subgroup of GG such that SS is a normal subgroup of it. Of course, if SS is itself a normal subgroup of GG, then its normalizer coincides with the whole of GG.

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. This solves the elementary problem of embedding a 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.

Examples

Revised on November 1, 2013 06:40:25 by Urs Schreiber (145.116.130.141)