Fusion systems on $p$-groups of sectional rank 3

Abstract

In this thesis we study saturated fusion systems on $p$-groups having sectional rank 3, for $p$ odd. We obtain a complete classification of simple fusion systems on p-groups having sectional rank 3 for $p$ ≥ 5, exhibiting a new simple exotic fusion system on a 7-group of order 7$^∧$5. We introduce the notion of pearls, defined as essential subgroups isomorphic to the groups C$_p$ X $_p$ and $p$$_+$$^1$$^+$$^2$ (for odd), and we illustrate some properties of fusion systems involving pearls. As for $p$ = 3, we determine the isomorphism type of a certain section of the 3-groups considered