Proving a conjecture for fusion systems on a class of groups

TL;DR

This paper proves that for a specific class of exceptional p-groups, exotic fusion systems and block-exotic fusion systems are equivalent, confirming a conjecture in this area of group theory.

Contribution

It establishes the conjecture for all fusion systems on exceptional p-groups of maximal nilpotency class for primes p ≥ 5, extending previous results and including new cases.

Findings

Exotic and block-exotic fusion systems coincide for these groups.

No exotic fusion systems on 2-groups of maximal class.

Block-exoticity of two specific exotic fusion systems at p=3.

Abstract

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional $p$-groups of maximal nilpotency class, where $p \geq 5$. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. 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. Considering small primes, there are no exotic fusion systems on $2$-groups of maximal class and for $p = 3$, we prove block-exoticity of two exotic fusion systems described by Diaz--Ruiz--Viruel.