Fusion Systems on a Sylow $p$-subgroup of $\mathrm{G}_2(p^n)$ or $\mathrm{PSU}_4(p^n)$

Abstract

For any prime p and S a p-group isomorphic to a Sylow p-subgroup of G 2 (p n ) or PSU 4 (p n ) with n ∈ N, we determine all saturated fusion systems supported on S up to isomorphism.

Exotic Fusion Systems Related to Sporadic Simple Groups

“…We remark that, perhaps aside from the Sylow 3-subgroup of Fi 22 , the remaining cases are large and complex enough that it is laborious and needlessly computationally expensive to verify any results using the fusion systems package in MAGMA [BCP97], [PS21]. Throughout this work, we lean on a small portion of these algorithms for the determination of the essentials subgroups of the saturated fusion systems under investigation (as in Proposition 3.3), although the techniques used in [PS18] and [vBe21b] could be employed here instead. We record that several of the main theorems have been verified using the full potential of this MAGMA package.…”