Algorithms for fusion systems with applications to $p$-groups of small order

Abstract: For a prime $p$, we describe a protocol for handling a specific type of fusion system on a $p$-group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the $p$-group. This allows a computational check of saturation. These procedures have been implemented using MAGMA. We describe a program to search for saturated fusion systems $\mathcal{F}$ on $p$-groups with $O_p(\mathcal{F})=1$ and $O^p(\mathcal{F})=\mathcal{F}$. Employing these computational methods we determine all such fusion system on groups of order $p^n$ where $(p,n) \in {(3,4),(3,5),(3,6),(3,7),(5,4),(5,5),(5,6),(7,4),(7,5)}$. This gives the first complete picture of which groups can support saturated fusion systems on small $p$-groups of odd order.