### Abstract

We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system F on a finite p-group S, and in the cases where p is odd or F is S-4-free, we show that Z(N-F(J(S))) = Z(F) (Glauberman) and that if C-F(Z(S)) = N-F(J(S)) = F-S(S), then F = F-S(S) (Thompson). As a corollary, we obtain a stronger form of Frobenius' theorem for fusion systems, applicable under the above assumptions and generalizing another result of Thompson.

Original language | English |
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Pages (from-to) | 495-503 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 2 |

Early online date | 17 Sep 2008 |

DOIs | |

Publication status | Published - Feb 2009 |

## Cite this

Diaz, A., Glesser, A., Mazza, N., & Park, S. (2009). Glauberman's and Thompson's theorems for fusion systems.

*Proceedings of the American Mathematical Society*,*137*(2), 495-503. https://doi.org/10.1090/S0002-9939-08-09690-1