Saturated Fusion Systems of Essential Rank

In this thesis we primarily consider saturated fusion systems which are of essential rank 1, i.e. which contain only one conjugacy class of essential subgroups. We prove that they all can be realized by an amalgam of two finite groups. This leads us to the application of amalgam related methods like pushing up techniques and results about FF-modules. As a first key result, in fusion systems of essential rank 1 satisfying certain minor extra conditions, we identify SL2(q) acting on a natural module inside the normalizer of an essential subgroup. Note that there still is no bound on the number of non-trivial chief factors inside an essential subgroup. This suggests that we need some stronger assumptions and motivates us to introduce minimal fusion systems. Here our definition is natural in the sense that every p-reduced fusion system contains a non-trivial section which is minimal. Expecting that the essential rank of a minimal fusion system is restricted, we again focus on the case of essential rank 1. We then obtain detailed information about the normalizer of an essential subgroup. If p = 2, this leads to a classification of the whole fusion system, showing that the arising examples are all non-exotic.