# Definition:Normalizer

## Definition

Let $G$ be a group.

Let $S$ be a subset of $G$.

Then the normalizer of $S$ in $G$ is the set $\map {N_G} S$ defined as:

$\map {N_G} S := \set {a \in G: S^a = S}$

where $S^a$ is the $G$-conjugate of $S$ by $a$.

If $S$ is a singleton such that $S = \set s$, we may also write $\map {N_G} s$ for $\map {N_G} S = \map {N_G} {\set s}$, as long as there is no possibility of confusion.

## Also denoted as

The notation $\map N {S; G}$ is sometimes seen for the normalizer of $S$ in $G$.

## Also see

• Results about normalizers can be found here.

## Linguistic Note

The UK English spelling of normalizer is normaliser.