# Definition:Conjugacy Class

## Definition

The equivalence classes into which the conjugacy relation divides its group into are called conjugacy classes.

The conjugacy class of an element $x \in G$ can be denoted $\conjclass x$.

## Also denoted as

Some authors use the notation $\operatorname {cl} \paren x$, but this can be confused with the notation for closure in the context of topology, so its use is not recommended.

Variants on the $\mathrm C$ motif can be seen: $C_x$ or $\map C x$ are fairly common.

## Examples

### Conjugacy Classes of Symmetric Group on $3$ Letters

Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:

$\begin{array}{c|cccccc} \circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$

The conjugacy classes of $S_3$ are:

 $\displaystyle$  $\displaystyle \set e$ $\displaystyle$  $\displaystyle \set {\tuple {123}, \tuple {132} }$ $\displaystyle$  $\displaystyle \set {\tuple {12}, \tuple {13}, \tuple {23} }$

## Also see

• Results about conjugacy classes can be found here.

## Technical Note

The $\LaTeX$ code for $\conjclass {x}$ is \conjclass {x} .

When the subscript is a single character, it is usual to omit the braces:

\conjclass x