# Difference between revisions of "Conjugacy class"

(stub) |
Duck master (talk | contribs) (added more info) |
||

Line 1: | Line 1: | ||

− | A '''conjugacy class''' is | + | A '''conjugacy class''' is a particular subset of a [[group]]. |

− | Let <math>G</math> be a group. Consider the action of <math>G</math> on itself by [[inner automorphism]]s. The [[orbit]]s of <math>G</math> are then called '''conjugacy classes'''. | + | Let <math>G</math> be a group. Consider the action of <math>G</math> on itself by [[inner automorphism]]s. The [[orbit]]s of <math>G</math> are then called '''conjugacy classes'''. By expanding the definition, it is easy to show that two elements <math>g</math> and <math>g'</math> are in the same conjugacy class iff there is an element <math>x</math> such that <math>g' = x^{-1}gx</math>. |

Two [[subset]]s <math>H</math> and <math>H'</math> of <math>G</math> are called ''conjugate'' if there exists <math>\alpha \in G</math> for which <math>H</math> is the image of <math>H'</math> under <math>\text{Int}(\alpha)</math>. | Two [[subset]]s <math>H</math> and <math>H'</math> of <math>G</math> are called ''conjugate'' if there exists <math>\alpha \in G</math> for which <math>H</math> is the image of <math>H'</math> under <math>\text{Int}(\alpha)</math>. | ||

+ | |||

+ | The [[characters|character]] of any group <math>G</math> are constant on conjugacy classes. | ||

{{stub}} | {{stub}} |

## Latest revision as of 12:00, 16 July 2020

A **conjugacy class** is a particular subset of a group.

Let be a group. Consider the action of on itself by inner automorphisms. The orbits of are then called **conjugacy classes**. By expanding the definition, it is easy to show that two elements and are in the same conjugacy class iff there is an element such that .

Two subsets and of are called *conjugate* if there exists for which is the image of under .

The character of any group are constant on conjugacy classes.

*This article is a stub. Help us out by expanding it.*