### 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

*Proceedings of the American Mathematical Society*,

*137*(2), 495-503. https://doi.org/10.1090/S0002-9939-08-09690-1

**Glauberman's and Thompson's theorems for fusion systems.** / Diaz, Antonio; Glesser, Adam; Mazza, Nadia; Park, Sejong.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 137, no. 2, pp. 495-503. https://doi.org/10.1090/S0002-9939-08-09690-1

}

TY - JOUR

T1 - Glauberman's and Thompson's theorems for fusion systems

AU - Diaz, Antonio

AU - Glesser, Adam

AU - Mazza, Nadia

AU - Park, Sejong

PY - 2009/2

Y1 - 2009/2

N2 - 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.

AB - 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.

U2 - 10.1090/S0002-9939-08-09690-1

DO - 10.1090/S0002-9939-08-09690-1

M3 - Article

VL - 137

SP - 495

EP - 503

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -