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

Abstract

For a prime p p , we describe a protocol for handling a specific type of fusion system on a p p -group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the p p -group. This allows a computational check of saturation. These procedures have been implemented using Magma. We describe a computer program which searches for saturated fusion systems F \mathcal {F} on p p -groups with O p ( F ) = 1 O_p(\mathcal {F})=1 and O p ( F ) = F O^p(\mathcal {F})=\mathcal {F} . Employing these computational methods we determine all such fusion systems on groups of order p n p^n where ( p , n ) ∈ { ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 3 , 7 ) , ( 5 , 4 ) , ( 5 , 5 ) , ( 5 , 6 ) , ( 7 , 4 ) , ( 7 , 5 ) } (p,n) \in \{(3,4),(3,5),(3,6),(3,7),(5,4),(5,5),(5,6),(7,4),(7,5)\} . This gives the first complete picture of which groups can support saturated fusion systems with O p ( F ) = 1 O_p(\mathcal {F})=1 and O p ( F ) = F O^p(\mathcal {F})=\mathcal {F} on small p p -groups of odd order.

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

“…In all of the examples they discovered, the 3-group has an abelian subgroup of index 3 [8]. In 2019, Parker and Semeraro, using their computational approach to saturated fusion systems [49] uncovered a saturated fusion system on a rank 2 group of order 3 6 which has maximal class and no abelian subgroups of index 3. This gave rise to the article [48].…”

“…In Section 14, we show that an automorphism group G of G 0 = PSL p (r ap ) which projects diagonally into a Sylow p-subgroups of Out(G 0 ) between the image of PGL p (r a ) and the image of a field automorphism of order p provides realizable examples of Theorem C (iii) with, for S ∈ Syl p (G), Out F S (G) (γ 1 (S)) ∼ = Sym(p). By [49,Theorem 6.2], the subfusion systems generated by the F S (G)-pearls gives an example of (i) in the case that γ 2 (S) is abelian. We also remark that in Theorem C (ii) when γ 1 (S)/Z(S) is abelian, we may apply Oliver's Theorem [42,Theorem 2.8] to see that O p (F /Z(S)) is simple and exotic.…”

Exotic Fusion Systems Related to Sporadic Simple Groups

“…We utilize MAGMA [PS21] to verify the following proposition, although there are certainly handwritten arguments which would lead to the same conclusion.…”

“…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.…”

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

“…As mentioned throughout, there is some exceptional behaviour for small values of p and n where the fusion systems of some other finite simple groups appear. In these instances, we generally appeal to a package in MAGMA [PS21] to determine a list of radical, centric subgroups and a list of saturated fusion systems.…”