Center of a Group - Imported Knowlpedia Content

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

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

It is a subgroup of $G$ .

Remarks

The center measures how far $G$ is from being abelian: $G$ is abelian iff $Z(G)=G$ . Moreover, $Z(G)$ is always a characteristic subgroup (hence normal), and it can be described as the intersection of all centralizers of 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 $\{\pm 1\}$ .