Conjugacy class - Imported Knowlpedia Content

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\}.

Equivalent characterizations

Equivalently, $\mathrm{Cl}_G(g)$ is the set of all conjugates of $g$ .

Remarks

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.