Two families of exotic fusion systems

Abstract

We construct two infinite families of exotic fusion systems.

Saturated fusion systems on $p$-groups of maximal class

“…These fusion systems, from their very construction are closely related to finite simple groups. Other exotic systems are constructed as fusion systems of free amalgamated products, such as those in [10,16,50] and these are often far away from being realized by finite groups in that the p-groups are often not closely related to the Sylow p-subgroup of a finite simple group. Many of these fusion systems are defined on p-groups of maximal class.…”

“…This gave rise to the article [48]. From a different direction, Clelland and Parker [16] constructed families of saturated fusion systems on a Sylow p-subgroup T of groups of shape q a : SL 2 (q) where 2 ≤ a ≤ p and q = p b . When a = 2, T is a Sylow p-subgroup of SL 3 (q) and, for a = 3, T is a Sylow p-subgroup of Sp 4 (q).…”

Exotic Fusion Systems Related to Sporadic Simple Groups

“…We remark that the set of essentials {E D 3 } in some ways behave similarly to pearls as defined in [Gra18], or the extensions of pearls as found in [Oli14]. In some ways, our class {E D 3 } motivates an examination of a generalization of pearls to q-pearls P where O p (G) ∼ = q 2 : SL 2 (q) for G some model of N F (P ) and q = p n , as in [CP10].…”

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

“…The work of Henke-Shpectorov [HS] and Clelland-Parker [CP10] shows that for p odd the Sylow p-subgroup of PSp 4 (p n ) supports exotic fusion systems, and so this restriction is required in Conjecture 2. All the other known sporadic groups or exotic fusion systems on such p-groups only occur for small values of p. For example, the conjecture holds if G = G 2 (p), PSU 4 (p), SL 3 (p n ) by [PS18,Mon18,Cle07].…”