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# Saturated fusion systems and finite groups

Clelland, Murray Robinson (2007)
Ph.D. thesis, University of Birmingham.

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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].

Type of Work: Ph.D. thesis. Parker, Christopher Schools (1998 to 2008) > School of Mathematics & Statistics Mathematics QA Mathematics University of Birmingham Check for printed version of this thesis 70
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