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eBook
On Fusion Systems of Component Type
2019
Overview
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.
Publisher
American Mathematical Society
Subject
ISBN
1470435209, 9781470435202
