Conjugacy Class of Element of Center is Singleton

Theorem

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.

Corollary

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

Proof

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

\(\ds a\)

\(\in\)

\(\ds \map Z G\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall x \in G: \, \)

\(\ds x a\)

\(=\)

\(\ds a x\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall x \in G: \, \)

\(\ds x a x^{-1}\)

\(=\)

\(\ds a\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \conjclass a\)

\(=\)

\(\ds \set a\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Exercise $25.16 \ \text{(a)}$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 51$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 48.3$ Conjugacy
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conjugacy class