# Definition:Centralizer

## Definition

### Centralizer of a Group Element

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 $a$.

### Centralizer of a Subset of a Group

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

Let $S \subseteq G$.

The centralizer of $S$ (in $G$) is the set of elements of $G$ which commute with all $s \in S$:

$\map {C_G} S = \set {x \in G: \forall s \in S: x \circ s = s \circ x}$

### Centralizer of a Subgroup

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

Let $H \le \struct {G, \circ}$.

The centralizer of $H$ (in $G$) is the set of elements of $G$ which commute with all $h \in H$:

$\map {C_G} H = \set {g \in G: \forall h \in H: g \circ h = h \circ g}$

### Centralizer of a Ring Subset

Let $S$ be a subset of a ring $\struct {R, +, \circ}$.

The centralizer of $S$ in $R$ is defined as:

$\map {C_R} S = \set {x \in R: \forall s \in S: s \circ x = x \circ s}$

That is, the centralizer of $S$ is the set of elements of $R$ which commute with all elements of $S$.

## Also denoted as

Some sources reverse the subscript and argument and write, for example:

$\map {C_a} G$

for the centralizer of $a$ in $G$.

## Linguistic Note

The UK English spelling of centralizer is centraliser.