Center is Intersection of Centralizers

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


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

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

by definition of set intersection.

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


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

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

Hence the result.