Saturated fusion systems over $2$-groups

Abstract

We develop methods for listing, for a given 2-group $S$, all nonconstrained centerfree saturated fusion systems over $S$. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order $2^7$ and two of order $2^{10}$. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order.