Definition:Centralizer

Centralizer of a Group Element
Let $\left({G, \circ}\right)$ be a group.

Let $a \in \left({G, \circ}\right)$.

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


 * $C_G \left({a}\right) = \left\{{x \in G: x \circ a = a \circ x}\right\}$

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

Centralizer of a Subgroup
Let $\left({G, \circ}\right)$ be a group.

Let $H \le \left({G, \circ}\right)$.

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


 * $C_G \left({H}\right) = \left\{{g \in G: \forall h \in H: g \circ h = h \circ g}\right\}$

Centralizer of a Ring Subset
Let $S$ be a subset of a ring $\left({R, +, \circ}\right)$.

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


 * $C_R \left({S}\right) = \left\{{x \in R: \forall s \in S: s \circ x = x \circ s}\right\}$

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

Group definition

 * : $\S 25$: Exercise $25.16 \ \text{(b)}$
 * : $\S 37 \ (1)$
 * : $\S 10$: Definition $10.24$

Ring definition

 * : $\S 21$