# conjugacy class

Two elements $g$ and ${g}^{\prime}$ of a group $G$ are said to be conjugate if there exists $h\in G$ such that ${g}^{\prime}=hg{h}^{-1}$. Conjugacy of elements is an equivalence relation^{}, and the equivalence classes^{} of $G$ are called conjugacy classes^{}.

Two subsets $S$ and $T$ of $G$ are said to be conjugate if there exists $g\in G$ such that

$$T=\{gs{g}^{-1}\mid s\in S\}\subset G.$$ |

In this situation, it is common to write $gS{g}^{-1}$ for $T$ to denote the fact that everything in $T$ has the form $gs{g}^{-1}$ for some $s\in S$. We say that two subgroups^{} of $G$ are conjugate if they are conjugate as subsets.

Title | conjugacy class |
---|---|

Canonical name | ConjugacyClass |

Date of creation | 2013-03-22 12:18:09 |

Last modified on | 2013-03-22 12:18:09 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | conjugate |

Synonym | conjugate set |

Synonym | conjugate subgroup |

Related topic | ConjugacyClassFormula |