# class equation

The conjugacy classes^{} of a group form a partition^{} of its elements.
In a finite group^{}, this means that the order of the group is
the sum of the number of elements of the distinct conjugacy classes.
For an element $g$ of group $G$,
we denote the centralizer^{} in $G$ of $g$ by ${C}_{G}(g)$.
The number of elements in the conjugacy class of $g$ is $[G:{C}_{G}(g)]$,
the index of ${C}_{G}(g)$ in $G$.
For an element $g$ of the center $Z(G)$ of $G$,
the conjugacy class of $g$ consists of the singleton $\{g\}$.
Putting this together gives us the *class equation ^{}*

$$|G|=|Z(G)|+\sum _{i=1}^{m}[G:{C}_{G}({x}_{i})]$$ |

where the ${x}_{i}$ are elements of the distinct conjugacy classes contained in $G\setminus Z(G)$.

Title | class equation |
---|---|

Canonical name | ClassEquation |

Date of creation | 2013-03-22 13:10:41 |

Last modified on | 2013-03-22 13:10:41 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 20E45 |

Synonym | conjugacy class formula |

Related topic | ConjugacyClass |