Center is Intersection of Centralizers

Theorem
The center of a group is the intersection of all the centralizers of that group:


 * $$Z \left({G}\right) = \bigcap_{x \in G} C_G \left({x}\right)$$

Proof
Follows directly from the definition:


 * $$Z \left({G}\right) = \left\{{x \in G: \forall g \in G: x g = g x}\right\}$$
 * $$C_G \left({x}\right) = \left\{{g \in G: x g = g x}\right\}$$