Conjugacy class

Let $G$ be a group and let $g\in G$. The conjugacy class of $g$ in $G$ is the subset $\mathrm{Cl}_G(g)=\{xgx^{-1} : x\in G\}.$ Equivalently, $\mathrm{Cl}_G(g)$ is the set of all conjugates of $g$.

Conjugacy classes are exactly the orbits of the conjugation action of $G$ on itself, so they form a partition of $G$. An element lies in the center of $G$ if and only if its conjugacy class is a singleton. The sizes of conjugacy classes feature prominently in the class equation .

Examples:

• In $S_3$, the conjugacy classes are $\{e\}$, the three transpositions $\{(12),(13),(23)\}$, and the two $3$-cycles $\{(123),(132)\}$.
• If $G$ is abelian, then $\mathrm{Cl}_G(g)=\{g\}$ for every $g\in G$.
• In $D_8=\langle r,s\mid r^4=s^2=e,\ srs=r^{-1}\rangle$, one has $\mathrm{Cl}_{D_8}(r)=\{r,r^{-1}\}=\{r,r^3\}$, so conjugacy classes need not be singletons in nonabelian groups.