Definition:Conjugacy Class

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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:

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


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


Sources