Must the center of a group be made up of congugacy classes?

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- May 13th 2011, 12:51 PMalexandrabel90Centers
Must the center of a group be made up of congugacy classes?

- May 13th 2011, 06:56 PMDrexel28
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 $\displaystyle \mathcal{Z}(G)\unlhd G$ for any group $\displaystyle G$ (since $\displaystyle \mathcal{Z}(G)=\ker\phi$ where $\displaystyle \phi:G\to \text{Aut}(G):g\mapsto i_g$ where $\displaystyle i_g$ is the inner automorphism) and normal subgroups are subgroups which are unions of conjugacy classes. Perhaps simpler is the fact that, by definition, $\displaystyle \displaystyle \mathcal{Z}(G)=\bigsqcup_{C_g: \text{card}(C_g)=1}C_g$ where $\displaystyle C_g$ is the conjugacy class of $\displaystyle g$ and the square union is just a fancy way of saying that the sets in the union are pairwise disjoint.