nLab normalizer

Definition

Given a subset $S$ of (the underlying set of) a group $G$, its normalizer $N(S) = N_G(S)$ is the subgroup of $G$ consisting of all elements $g\in G$ such that $g S = S g$, i.e. for each $s\in S$ there is $s'\in S$ such that $g s = s'g$.

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

Properties

Normalization and Weyl group

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

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

Hence when $S$ is a group then the quotient

$$W_G S \coloneqq N_G(S)/S$$

is a quotient group. This is also called the Weyl group of $S$ in $G$. (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 $G$ 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. This solves the elementary problem of embedding a 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 $K$.

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 $U \stackrel{u}{\to} N \stackrel{f}{\to}T$ of a monomorphism $U \to T$ with $u$ a normal monomorphism.

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

Related concepts

• stabilizer subgroup

• centralizer subgroup

References

• Glen Bredon, Section 0.1 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)

• Peter May, Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

In semiabelian categories:

• James Richard Andrew Gray, Normalizers, centralizers and action representability in semiabelian categories, Applied Categorical Structures 22(5-6), 981–1007, 2014.

• Dominique Bourn, James Richard Andrew Gray, Normalizers and split extensions (arXiv:1307.4845)