Proving a conjecture for fusion systems on a class of groups

Patrick Serwene

Published 2023-02-10Version 1

We prove that the conjecture that exotic and block-exotic fusion systems coincide holds all for all fusion systems on p-groups of maximal nilpotency class, where p is either a small prime or $p \geq 5$ and the group is also exceptional. For $p = 3$, this is achieved by considering exotic fusion systems described by Diaz--Ruiz--Viruel. For $p \geq 5$, this is achieved by proving a family of exotic fusion systems discovered by Parker and Stroth is also block-exotic. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups.