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Full Description
Let $p$ be a prime and$S$ a finite $p$-group. A $p$-fusion system on $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are certain injective group homomorphisms. Fusion systems are of interest in modular representation theory, algebraic topology, and local finite group theory. The book provides a characterization of the 2-fusion systems of the groups of Lie type and odd characteristic, a result analogous to the Classical Involution Theorem for groups. The theorem is the most difficult step in a two-part program. The first part of the program aims to determine a large subclass of the class of simple 2-fusion systems, while part two seeks to use the result on fusion systems to simplify the proof of the theorem classifying the finite simple groups.
Contents
Background and overview: Introduction
The major theorems and some background
Basics and examples:
Some basic results
Results on $\tau$
$W(\tau)$ and $M(\tau)$
Some examples
Theorems 2 through 5: Theorems 2 and 4
Theorems 3 and 5
Coconnectedness: $\tau^{\circ}$ not coconnected
Theorem 6: $\Omega =\Omega(z)$ of order 2
$\vert\Omega(z)\vert>2$
Some results on generation
$\vert\Omega(z)\vert=2$ and the proof of Theorem 6
Theorems 7 and 8: $\vert\Omega(z)\vert=1$ and $\mu$ abelian
More generation
$\vert\Omega(z)\vert=1$ and $\mu$ nonabelian
Theorem 1 and the Main Theorem: Proofs of four theorems
References and Index: Bibliography
Index.
