# nLab normalizer

group theory

### Cohomology and Extensions

Given a subset $S$ of a group $G$, its normalizer $N\left(S\right)={N}_{G}\left(S\right)$ is the subgroup of $G$ consisting of all elements $g\in G$ such that $gS=Sg$, i.e. for each $s\in S$ there is $s\prime \in S$ such that $gs=s\prime g$. If $S$ is itself a subgroup, then $S$ is a normal subgroup of ${N}_{G}\left(S\right)$; moreover ${N}_{G}\left(S\right)$ is the largest subgroup of $G$ such that $S$ is a normal subgroup of it. Of course, if $S$ is itself a normal subgroup of $G$, then its normalizer coincides with the whole of $G$.

Each group $G$ embeds into the symmetric group $\mathrm{Sym}\left(G\right)$ on the underlying set of $G$ by the left regular representation $g↦{l}_{g}$ where ${l}_{g}\left(h\right)=gh$. 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 $\mathrm{Sym}\left(G\right)$ 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 $K$.

Revised on March 11, 2013 15:02:29 by Andrew Stacey (92.21.167.146)