Definition:Conjugacy Class

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This page is about Conjugacy Class in the context of Group Theory. For other uses, see 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:

\(\ds \) \(\) \(\ds \set e\)
\(\ds \) \(\) \(\ds \set {\tuple {123}, \tuple {132} }\)
\(\ds \) \(\) \(\ds \set {\tuple {12}, \tuple {13}, \tuple {23} }\)


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


Sources

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