group theory

# Contents

## Definition

Given a subset $S$ 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 $g s=s'g$.

If $S$ is itself a subgroup, then $S$ is a normal subgroup of $N_G(S)$; moreover $N_G(S)$ 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 $Sym(G)$ 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 $K$.

## Examples

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