# Centralizer of Subset is Intersection of Centralizers of Elements

## Theorem

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

Let $S \subseteq G$.

Let $\map {C_G} S$ be the centralizer of $S$ in $G$.

Then:

$\ds \map {C_G} S = \bigcap_{x \mathop \in S} \map {C_G} x$

where $\map {C_G} z$ is the centralizer of $x$ in $G$.

## Proof

 $\ds \map {C_G} S$ $=$ $\ds \set {g \in G: \forall x \in S: g \circ x = x \circ g}$ Definition of Centralizer of Group Subset $\ds$ $=$ $\ds \set {g \in G: \forall x \in S: g \in \map {C_G} x}$ Definition of Centralizer of Group Element $\ds$ $=$ $\ds \bigcap_{x \mathop \in S} \map {C_G} x$ Definition of Intersection of Family

$\blacksquare$