We study a problem in fusion systems designed to mimic, simplify, and generalize parts of the Classification of Finite Simple Groups. A 2-fusion system is a 2-group S with a family of injective homomorphisms on subgroups of S. Fusion systems arise in the study of modular representations and classifying spaces, and our results have potential ramifications beyond finite group theory. One problem is to determine S or, whenever possible, the entire 2-fusion system from the knowledge of certain subgroups and maps. We consider the case where S contains a subgroup and maps that arise in the Classification with standard component of type SL 2 q .One of the goals in the study of fusion systems is to improve and clarify the Classification of the Finite Simple Groups by first proving results for fusion systems and then applying those results to groups. In the Classification, the simple groups are split between those of characteristic 2-type and those of component type. Following a major program of research laid out in Aschbacher et al.[4], we work toward a classification of "fusion systems of component type" in order to work toward a new proof of the Classification for groups of component type. To that end, this paper describes the classification of the simple, saturated 2-fusion systems in a "small case" of the Classical Involution Theorem [1, 2] by considering fusion systems over a finite 2-group S possessing a weakly closed (generalized) quaternion subgroup R which is also strongly closed in the centralizer of its unique involution. We will assume further that Q = C S R is "tightly embedded" in . This is the fusion system version of a standard component of type SL 2 q , q odd. For another instance of work in support of this program, see the work of Lynd [11].
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