Center of a Group

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 $G$.

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 $\{\pm 1\}$.