nLab normalizer




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.


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.)



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.


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.


  • 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)

Last revised on September 5, 2021 at 07:55:34. See the history of this page for a list of all contributions to it.