Conjugacy Class of Element of Center is Singleton

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Let $G$ be a group.

Let $\map Z G$ denote the center of $G$.

The elements of $\map Z G$ form singleton conjugacy classes, and the elements of $G \setminus \map Z G$ belong to multi-element conjugacy classes.


The number of single-element conjugacy classes of $G$ is the order of $\map Z G$ and divides $\order G$.


Let $\conjclass a$ be the conjugacy class of $a$ in $G$.

\(\ds a\) \(\in\) \(\ds \map Z G\)
\(\ds \leadstoandfrom \ \ \) \(\ds \forall x \in G: \, \) \(\ds x a\) \(=\) \(\ds a x\)
\(\ds \leadstoandfrom \ \ \) \(\ds \forall x \in G: \, \) \(\ds x a x^{-1}\) \(=\) \(\ds a\)
\(\ds \leadstoandfrom \ \ \) \(\ds \conjclass a\) \(=\) \(\ds \set a\)