# Centers

What does this mean? Is it true that the center of a group is a union of conjugacy classes? The answer is yes for two reasons. The first reason is the fact that $\mathcal{Z}(G)\unlhd G$ for any group $G$ (since $\mathcal{Z}(G)=\ker\phi$ where $\phi:G\to \text{Aut}(G):g\mapsto i_g$ where $i_g$ is the inner automorphism) and normal subgroups are subgroups which are unions of conjugacy classes. Perhaps simpler is the fact that, by definition, $\displaystyle \mathcal{Z}(G)=\bigsqcup_{C_g: \text{card}(C_g)=1}C_g$ where $C_g$ is the conjugacy class of $g$ and the square union is just a fancy way of saying that the sets in the union are pairwise disjoint.