Center of a Group - Knowlpedia

Let $G$ be a group. The center of $G$ is the subgroup

Z(G)=\{z\in G:zx=xz\text{ for every }x\in G\}.

Remarks

The group $G$ is abelian if and only if $Z(G)=G$ . Moreover, $Z(G)$ is always a characteristic subgroup (hence normal), and it is the intersection of the centralizers of all elements of $G$ .

Examples

If $G$ is abelian, then $Z(G)=G$ .
In $S_3$ , $Z(S_3)=\{e\}$ .
In the quaternion group $Q_8$ , the center is $\{\pm 1\}$ .