turkmath.org

The sectional rank of a finite p-group is the largest rank of any abelian subquotient. The sectional p-rank of a finite group is the sectional rank of its Sylow p-subgroups. I will describe a result listing all reduced, indecomposable fusion systems over 2-groups of sectional rank at most four. This is motivated by a theorem of Gorenstein and Harada, where they listed all finite simple groups of sectional 2-rank at most four. The new result contains no surprises: the fusion systems in question are all those of simple groups on the Gorenstein-Harada list. But the method of proof seems very different, since it is based on studying the different types of essential subgroups which can occur, rather than the centralizers of involutions. This also leads to a different way of organizing the final result, which I hope will be of interest.