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On Fusion Systems of Component Type
About this Title
Michael Aschbacher, California Institute of Technology, Pasadena, California 91125
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 257, Number 1236
ISBNs: 978-1-4704-3520-2 (print); 978-1-4704-4955-1 (online)
DOI: https://doi.org/10.1090/memo/1236
Published electronically: January 4, 2019
MSC: Primary 20D05; Secondary 55R35
Table of Contents
Chapters
- Introduction
- 1. Preliminaries
- 2. Some Lemmas on Fusion Systems
- 3. Tight embedding
- 4. More on tight embedding
- 5. Split extensions
- 6. Component combinatorics
- 7. The proof of Theorem
- 8. Terminal components
- 9. Standard subsystems
Abstract
This memoir begins a program to classify a large subclass of the class of simple saturated 2-fusion systems of component type. Such a classification would be of great interest in its own right, but in addition it should lead to a significant simplification of the proof of the theorem classifying the finite simple groups.
Why should such a simplification be possible? Part of the answer lies in the fact that there are advantages to be gained by working with fusion systems rather than groups. In particular one can hope to avoid a proof of the B-Conjecture, a important but difficult result in finite group theory, established only with great effort.
But in addition, the program involves a reorganization of the treatment of “groups of component type”, or perhaps more accurately, of “fusion systems of component type”. The groups of component type should be viewed as “odd” groups, in that most examples are groups of Lie type over fields of odd order. The remaining simple groups should be viewed as “even” groups, since most of the examples in this class are of Lie type over fields of even order. There are corresponding notions of “odd” and “even” 2-fusion systems.
In our program the class of odd groups, and/or fusion systems, is contracted in a carefully chosen manner, so as to avoid difficulties associated to certain “standard form problems”. This has the effect of greatly simplifying the treatment of the odd 2-fusion systems, and then also the treatment of the odd simple groups. Of course the flip side of such a reorganization is to enlarge the class of even objects, so that the approach may make it more difficult to treat that class. But it is our sense that the trade off should lead to a net simplification.
This change in the partition of simple groups into odd and even groups is not dissimilar to the one in the program of Gorenstein, Lyons, and Solomon (hereafter referred to as GLS) to rewrite the proof of the classification.
In the introduction, we expand upon these themes, making them a bit more precise, supplying some background, and eventually stating some of our major theorems. Then in the body of the paper, we fill in details and begin the actual program.
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