Centralizer of Subset is Intersection of Centralizers of Elements

Theorem

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

Let $S \subseteq G$.

Let $C_G \left({S}\right)$ be the centralizer of $S$ in $G$.

Then:

$\displaystyle C_G \left({S}\right) = \bigcap_{x \mathop \in S} C_G \left({x}\right)$

where $C_G \left({x}\right)$ is the centralizer of $x$ in $G$.

Proof

By definition of the centralizer of $S$ in $G$:

$C_G \left({S}\right) = \left\{{g \in G: \forall s \in S: x \circ s = s \circ x}\right\}$