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Normalizers and centralizers of subnormal subsystems of fusion systems

Ellen Henke

TL;DR

This work develops a cohesive framework to study normalizers and centralizers of subnormal subsystems in saturated fusion systems by leveraging regular localities. It defines regular normalizers and centralizers at the locality level and translates these into well-defined normalizers and centralizers at the fusion-system level, including product constructions ER and Frattini-type decompositions. Central contributions include the creation of N_F(E) and C_F(E) with robust structural properties, criteria for saturation, and concrete descriptions and conjugation behavior, together with a detailed examination of conjugates and centralizers. The results unify and extend prior notions (e.g., Aschbacher, Oliver) and offer tools for analyzing subnormal pieces within fusion systems, with potential applications to classifying finite groups through locality methods. The paper also provides explicit examples illustrating limitations of certain equivalences and the necessity of conditions like full normalization or centralization in the theory.

Abstract

Every saturated fusion system corresponds to a group-like structure called a regular locality. In this paper we study (suitably defined) normalizers and centralizers of partial subnormal subgroups of regular localities. This leads to a reasonable notion of normalizers and centralizers of subnormal subsystems of fusion systems.

Normalizers and centralizers of subnormal subsystems of fusion systems

TL;DR

This work develops a cohesive framework to study normalizers and centralizers of subnormal subsystems in saturated fusion systems by leveraging regular localities. It defines regular normalizers and centralizers at the locality level and translates these into well-defined normalizers and centralizers at the fusion-system level, including product constructions ER and Frattini-type decompositions. Central contributions include the creation of N_F(E) and C_F(E) with robust structural properties, criteria for saturation, and concrete descriptions and conjugation behavior, together with a detailed examination of conjugates and centralizers. The results unify and extend prior notions (e.g., Aschbacher, Oliver) and offer tools for analyzing subnormal pieces within fusion systems, with potential applications to classifying finite groups through locality methods. The paper also provides explicit examples illustrating limitations of certain equivalences and the necessity of conditions like full normalization or centralization in the theory.

Abstract

Every saturated fusion system corresponds to a group-like structure called a regular locality. In this paper we study (suitably defined) normalizers and centralizers of partial subnormal subgroups of regular localities. This leads to a reasonable notion of normalizers and centralizers of subnormal subsystems of fusion systems.

Paper Structure

This paper contains 18 sections, 59 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Some background
  3. Normalizers and centralizers in regular localities
  4. Normalizers and centralizers of subnormal subsystems
  5. Preliminaries
  6. Invariant subsystems
  7. Commuting Subsystems
  8. Two lemmas on fusion systems
  9. Products of subnormal subsystems with $p$-groups
  10. Some lemmas on partial groups and localities
  11. Some group theoretic lemmas
  12. Normalizers and centralizers
  13. Some basic results
  14. The regular normalizer and the centralizer of $\mathcal{H}$ in $\mathcal{L}$
  15. The normalizer of $\mathcal{E}$
  16. ...and 3 more sections

Key Result

Theorem A

Let $(\mathcal{L},\Delta,S)$ be a regular locality over $\mathcal{F}$ and $\mathcal{H}\unlhd\!\unlhd\;\mathcal{L}$. Then the regular normalizer $\mathbb{N}_\mathcal{L}(\mathcal{H})$ is a partial subgroup of $\mathcal{L}$. Moreover, setting the following hold for every $S_0\in\operatorname{Syl}_p(N_G(\mathcal{H}))$:

Theorems & Definitions (132)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem A
  • Definition 4
  • Theorem B
  • Definition 6
  • Example 1
  • Theorem C
  • Theorem D
  • ...and 122 more