nLab
normalizer

Contents

Definition

Given a subset SS of (the underlying set 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.

Notice the similarity but also the difference to the definition of the centralizer subgroup, for which s=ss' = s in the above.

Properties

Normalization and Weyl group

If the subset SS is in fact a subgroup of GG, then it is a normal subgroup of the normalizer N G(S)N_G(S); and N G(S)N_G(S) is the largest subgroup of GG such that SS is a normal subgroup of it, whence the terminology normalizer.

Indeed, if SS is already a normal subgroup of GG, then its normalizer coincides with the whole of GG, and only then (e.g. here).

Hence when SS is a group then the quotient

W GSN G(S)/S W_G S \coloneqq N_G(S)/S

is a quotient group. This is also called the Weyl group of SS in GG. (This use of terminology is common in equivariant stable homotopy theory – see e.g. May 96, p. 13 – but not otherwise.)

Examples

Holomorph

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.

Generalization

In (Gray 14) the concept of the normalizer of a subgroup of a group is generalized to the normalizer of a monomorphism in any pointed category in terms of a universal decomposition UuNfTU \stackrel{u}{\to} N \stackrel{f}{\to}T of a monomorphism UTU \to T with uu a normal monomorphism.

In (Bourn-Gray 13) the condition that ww be a monomorphism is dropped.

References

In semiabelian categories: