Abstract

Fusion systems are structures that encode the properties of conjugation between $p$-subgroups of a group, for $p$ any prime number. It is always possible to define the saturated fusion system realized by a finite group on one of its Sylow $p$-subgroups. However, not all saturated fusion systems can be realized in this way: when this is the case, we say that the fusion system is exotic. The understanding of the behavior of exotic fusion systems (in particular at odd primes) is still an important open problem. It turns out that many of the known exotic fusion systems are defined on $p$-groups of maximal nilpotency class. In this talk we will present the classification of saturated fusion systems on $p$-groups of maximal nilpotency class, a joint work with C. Parker.Some of the techniques involved in the proof were developed during the first months of the GRA Programme in 2020.