Center is Intersection of Centralizers

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


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

Proof

 * Let $$x \in Z \left({G}\right)$$.

Then:
 * $$\forall g \in G: x g = g x$$

by definition of the center.

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

by definition of the centralizer.

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

by definition of set intersection.

So:
 * $$Z \left({G}\right) \subseteq \bigcap_{g \in G} C_G \left({g}\right)$$


 * Now let $$x \in \bigcap_{g \in G} C_G \left({g}\right)$$.

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

by definition of set intersection.

That is:
 * $$\forall g \in G: x g = g x$$

by definition of the centralizer.

So:
 * $$x \in \left\{{x \in G: \forall g \in G: x g = g x}\right\}$$

By definition of the center:
 * $$x \in Z \left({G}\right)$$.

So:
 * $$\bigcap_{g \in G} C_G \left({g}\right) \subseteq Z \left({G}\right)$$.

Hence the result.