# Frobenius group

A permutation group^{} $G$ on a set $X$ is Frobenius if no non-trivial element of $G$ fixes more than
one element of $X$. Generally, one also makes the restriction^{} that at least one non-trivial element
fix a point. In this case the Frobenius group is called non-regular.

The stabilizer^{} of any point in $X$ is called a Frobenius complement, and has the remarkable
property that it is distinct from any conjugate by an element not in the subgroup^{}. Conversely,
if any finite group^{} $G$ has such a subgroup, then the action on cosets of that subgroup makes
$G$ into a Frobenius group.

Title | Frobenius group |
---|---|

Canonical name | FrobeniusGroup |

Date of creation | 2013-03-22 13:16:30 |

Last modified on | 2013-03-22 13:16:30 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 20B99 |

Defines | Frobenius complement |