Conjugacy Classes of Center Elements are Singletons

Theorem

Let $G$ be a group.

Let $Z \left({G}\right)$ be the center of $G$.

The elements of $Z \left({G}\right)$ form singleton conjugacy classes, and the elements of $G \setminus Z \left({G}\right)$ belong to multi-element conjugacy classes.

Corollary

The number of single-element conjugacy classes of $G$ is the order of $Z \left({G}\right)$ and divides $G$.

Proof

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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a\)	\(\in\)	\(\displaystyle Z \left({G}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \forall x \in G: x a\)	\(=\)	\(\displaystyle a x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \forall x \in G: x a x^{-1}\)	\(=\)	\(\displaystyle a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \mathrm C_a\)	\(=\)	\(\displaystyle \left\{ {a}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $25.16 \ \text{(a)}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 51$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 48.3$