Glauberman’s and Thompson’s theorems for fusion systems

Abstract

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.

On Thompson’s p-complement theorems for saturated fusion systems

“…These results contribute to the list of nilpotency criteria for saturated fusion systems that generalize classical criteria for finite groups, and fits within the framework of previous work by Kessar-Linckelmann [9], Díaz-Glesser-Mazza-Park [6], Díaz-Glesser-Park-Stancu [7], Cantarero-Scherer-Viruel [3] and Craven [5]. Indeed, Theorem 1.1 can also be deduced from [6,Corollary 4.6], although the proof of Thompson's pnilpotence criterion in [6] resorts to the group case, while the proof presented here is purely fusion theoretical. Another independent fusion theoretical proof of the odd prime case in Theorem 1.2 can be found in [5] This note is organized as follows.…”

Control of transfer and weak closure in fusion systems

“…In this paper, following the strategy of [8] as in our previous work [3], we generalize these results of Glauberman to arbitrary fusion systems: Theorem 1.1. K ∞ and K ∞ control weak closure of elements in every fusion system on a finite p-group when p 3.…”

“…We end this article with a recap in Appendix A on Glauberman's K ∞ and K ∞ constructions. Our general terminology follows [8] and [3]; in particular, by a fusion system we always mean a saturated fusion system.…”

Fusion systems

“…We also refer to [DGMP1,DGMP2] for examples of how it can be used when generalizing theorems about groups to theorems about fusion systems. …”