Saturated Fusion Systems

Sammanfattning: A fusion system is a category on a nite p-group, P, which encodes "conjugacy" relations among the subgroups of P. In this thesis fusion systems of nite groups and ways to prove saturation of abstract fusion systems is investigated. First an introduction to fusion systems of nite groups and the notion of abstract fusion systems is given. Theorems of Burnside and Frobenius regarding fusion systems of nite groups are considered and proven. Alperins fusion theorem formulated for nite groups is considered and used in the proofs. It is proved that all fusion systems of nite groups are saturated. An investigation in simpler ways of proving that a fusion system is saturated is done. First Alperins fusion theorem formulated for abstract fusion systems is considered which says that, a fusion system, denoted by F, is generated by the automorphism groups of some special subgroups.  Further investigation is done in how this set of special groups, that generates F, can be used to check if F is saturated. A theorem of Craven, though originally stated by Puig, is then considered and proven. The theorem says that is suffices to check that the conjugacy classes of, so called, F-centric subgroups are saturated in order to check that the fusion system F is saturated. Also a theorem of [5] is considered and proven. The theorem says that an even smaller set of conjugacy classes than the set of F-centric subgroups is needed to check saturation. Section 1-2 are written together with Karl Amundsson and Eric Ahlqvist.