# Normalizer of a subgroup

(Redirected from Normalizer)
This article defines a subgroup operator related to the subgroup property normal subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

For the associated subgroup property, refer normalizer subgroup

You might be looking for the more general notion of: normalizer of a subset of a group

## Definition

### Symbol-free definition

The normalizer (normaliser in British English) of a subgroup in a group is any of the following equivalent things:

1. The largest intermediate subgroup in which the given subgroup is normal.
2. The set of all elements in the group for which the induced inner automorphism restricts to an automorphism of the subgroup.
3. The set of all elements in the group that commute with the subgroup.

### Definition with symbols

The normalizer of a subgroup $H$ in a group $G$, denoted as $N_G(H)$, is defined as any of the following equivalent things:

1. The largest group $K$ for which $H \le K \le G$ and $H$ is normal in $K$.
2. The set of all elements $x$ for which the map sending $g$ to $xgx^{-1}$ restricts to an automorphism of $H$.
3. The set of all elements $x$ for which $Hx = xH$.

## Related subgroup properties

### Inverse image of whole group

A subgroup is normal in the whole group if and only if its normalizer is the whole group. Thus the collection of normal subgroups can be thought of as the inverse image of the whole group under the normalizer map.

### Iteration

The $k$-times iteration of normalizer is termed the $k$-hypernormalizer and a subgroup whose $k$-times hypernormalizer is the whole group is termed a $k$-hypernormalized subgroup. The condition of being $k$-hypernormalized is stronger than the condition of being $k$-subnormal.

### Fixed-points

A subgroup of a group that is its own normalizer is termed a self-normalizing subgroup.