Saturated fusion systems and finite groups

Clelland, Murray Robinson (2007). Saturated fusion systems and finite groups. University of Birmingham. Ph.D.

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Abstract

This thesis is primarily concerned with saturated fusion systems over groups of shape q $^r$ : q where q = p $^n$ for some odd prime p and some natural number n. We shall present two results related to these fusion systems. Our first result is a complete classification of saturated fusion systems over a Sylow p-subgroup of SL $_3$ (q) (which has shape q $^3$ : q). This extends a result of Albert Ruiz and Antonio Viruel, who studied the case when q = p in [36]. As an immediate consequence of this result we shall have a complete classification of p-local finite groups over Sylow p-subgroups of SL $_3$ (q). In the second half of this thesis we shall construct an infinite family of exotic fusion systems over some groups of shape p $^r$ : p. This extends some work of Broto, Levi and Oliver, who studied the case when r = 3 in [12].