Center of Group is Subgroup/Proof 2

Theorem

Let $G$ be a group.

The center $\map Z G$ of $G$ is a subgroup of $G$.

Proof

We have the result Center is Intersection of Centralizers.

That is, $\map Z G$ is the intersection of all the centralizers of $G$.

All of these are subgroups of $G$ by Centralizer of Group Element is Subgroup.

Thus from Intersection of Subgroups is Subgroup, $\map Z G$ is also a subgroup of $G$.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 37.2$ Some important general examples of subgroups