Glauberman’s and Thompson’s theorems for fusion systems

Díaz, Antonio; Glesser, Adam; Mazza, Nadia; Park, Sejong

doi:10.1090/S0002-9939-08-09690-1

Glauberman’s and Thompson’s theorems for fusion systems
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by Antonio Díaz, Adam Glesser, Nadia Mazza and Sejong Park PDF

Proc. Amer. Math. Soc. 137 (2009), 495-503 Request permission

Abstract:

We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system $\mathcal {F}$ on a finite $p$-group $S$, and in the cases where $p$ is odd or $\mathcal {F}$ is $S_4$-free, we show that $\mathrm {Z}(\mathrm {N}_{\mathcal {F}}(\mathrm {J}(S))) =\mathrm {Z}(\mathcal {F})$ (Glauberman) and that if $\mathrm {C}_{\mathcal {F}} (\mathrm {Z}(S))=\mathrm {N}_{\mathcal {F}}(\mathrm {J}(S))=\mathcal {F}_S(S)$, then $\mathcal {F}=\mathcal {F}_S(S)$ (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.

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