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Criteria for nilpotency of fusion systems

Jie Jian, Jun Liao, Heguo Liu

Abstract

Let $p$ be an odd prime and let $\mathcal{F}$ be a fusion system over a finite $p$-group $P$. A fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. In this paper we provide new criteria for saturated fusion systems $\mathcal{F}$ to be nilpotent, which can be viewed as extension of the $p$-nilpotency theorem of Glauberman and Thompson for fusion systems attributed to Kessar and Linckelmann.

Criteria for nilpotency of fusion systems

Abstract

Let be an odd prime and let be a fusion system over a finite -group . A fusion system is said to be nilpotent if . In this paper we provide new criteria for saturated fusion systems to be nilpotent, which can be viewed as extension of the -nilpotency theorem of Glauberman and Thompson for fusion systems attributed to Kessar and Linckelmann.
Paper Structure (4 sections, 14 theorems, 1 equation)

This paper contains 4 sections, 14 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Background on fusion systems and finite groups
  3. Preliminary Lemmas
  4. Proof of Theorem \ref{['main']} and Theorem \ref{['main2']}

Key Result

Theorem A

Let $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $P$, where $p$ is an odd prime. Then $\mathcal{F}=\mathcal{F}_{P}(P)$ if and only if $N_{\mathcal{F}}(I_{\mathcal{A}})=\mathcal{F}_{P}(P)$, where $\mathcal{A}\subseteq \mathfrak{Ab}(P)$ satisfies the following properties:

Theorems & Definitions (23)

  • Theorem A
  • Theorem B
  • Definition 2.1: AKO
  • Theorem 2.2: Glesser
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: KS
  • Lemma 2.6: KS
  • Theorem 2.7: Isaacs
  • Theorem 2.8: Frobenius, Isaacs
  • ...and 13 more