Reduced fusion systems over $p$-groups with abelian subgroup of index $p$: III

Abstract

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all F p G-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [CrOS] using the classification of finite simple groups.