Def/Center (group)

 =Properties= The center $Z(G)$ of any group $G$ is a subgroup of $G$. This can be checked as follows:

In fact, $Z(G)$ is a normal subgroup of $G$. We have seen above that $Z(G)$ is a subgroup. If $g \in Z(G)$, then every conjugate of $g$ is equal to $g$. Hence $Z(G)$ is normal.

A group $G$ is abelian if and only if $G = Z(G)$.