Print

Welz, Matthew

Mathematics

2012

PhD

In this thesis we study two problems in the area of fusion systems which are designed to mimic, simplify, and generalize parts of the Classification of Finite Simple Groups. In general, a finite simple group G is determined to a great extent by the structure and conjugacy pattern of a Sylow 2-subgroup. A 2-fusion system considers only a 2-group S equipped with a family of injective homomorphisms (called fusion maps) on subgroups of S without reference to aI) ambient group G. The general framework of fusion systems also arises naturally in the study of modular representations and classifying spaces; and so results proved for fusion systems have potential ramifications beyond the realm of finite group theory. One problem in this area is to determine S or, whenever possible, the entire 2-fusion system only from the knowledge of certain subgroups and fusion maps between these subgroups. In this thesis we consider two such problems: where S contains subgroups and fusion maps that arise in the Classification with standard components of type SL₂(q) and PS L₂(q). In particular, we give a characterization of simple, saturated fusion systems containing such components.