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Glauberman and Thompson's theorems for fusion systems.

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<mark>Journal publication date</mark>2009
<mark>Journal</mark>Proceedings of the American Mathematical Society
Volume137
Number of pages15
Pages (from-to)495-509
<mark>State</mark>Published
<mark>Original language</mark>English

Abstract

We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system $\CF$ on a finite $p$-group $S$, and in the cases where $p$ is odd or $\CF$ is $S_4$-free, we show that $\Z(\N_\CF(\J(S)))=\Z(\CF)$ (Glauberman), and that if $\C_\CF(\Z(S))=\N_\CF(\J(S))=\CF_S(S)$, then $\CF=\CF_S(S)$ (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.

Bibliographic note

First published in Proceedings of the American Mathematical Society in 137, 2009, published by the American Mathematical Society. Copyright 2008, American Mathematical Society