# 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 $G$:

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

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Exercise $(11)$