# Definition:Centralizer/Group Element

## Definition

Let $\struct {G, \circ}$ be a group.

Let $a \in \struct {G, \circ}$.

The centralizer of $a$ (in $G$) is defined as:

$\map {C_G} a = \set {x \in G: x \circ a = a \circ x}$

That is, the centralizer of $a$ is the set of elements of $G$ which commute with $a$.

## Also known as

Some sources call this the normalizer of $a$ in $G$ but that term generally has another meaning.

## Also see

• Results about centralizers can be found here.

## Linguistic Note

The UK English spelling of centralizer is centraliser.