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 $G$.
Also see
Linguistic Note
The UK English spelling of centralizer is centraliser.