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The Generalized Fitting Subsystem of a Fusion System
Michael Aschbacher, California Institute of Technology, Pasadena, CA

Memoirs of the American Mathematical Society
2011; 110 pp; softcover
Volume: 209
ISBN-10: 0-8218-5303-1
ISBN-13: 978-0-8218-5303-0
List Price: US$74
Individual Members: US$44.40
Institutional Members: US$59.20
Order Code: MEMO/209/986
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The notion of a fusion system was first defined and explored by Puig, in the context of modular representation theory. Later, Broto, Levi, and Oliver extended the theory and used it as a tool in homotopy theory. The author seeks to build a local theory of fusion systems, analogous to the local theory of finite groups, involving normal subsystems and factor systems. Among other results, he defines the notion of a simple system, the generalized Fitting subsystem of a fusion system, and prove the L-balance theorem of Gorenstein and Walter for fusion systems. He defines a notion of composition series and composition factors and proves a Jordon-Hölder theorem for fusion systems.

Table of Contents

  • Introduction
  • Background
  • Direct products
  • \(\mathcal {E}_{1}\wedge \mathcal {E}_{2}\)
  • The product of strongly closed subgroups
  • Pairs of commuting strongly closed subgroups
  • Centralizers
  • Characteristic and subnormal subsystems
  • \(T\mathcal {F}_{0}\)
  • Components
  • Balance
  • The fundamental group of \(\mathcal{F}^{c}\)
  • Factorizing morphisms
  • Composition series
  • Constrained systems
  • Solvable fusion systems
  • Fusion systems in simple groups
  • An example
  • Bibliography
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