Reduced fusion systems over 2-groups of small order

Abstract

Abstract. We prove, when S is a 2-group of order at most 2 9 , that each reduced fusion system over S is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a 2-group of order at most 2 9 is realizable. What is most interesting about this result is the method of proof: we show that among 2-groups with order in this range, the ones which can be Sylow 2-subgroups of finite simple groups are almost completely determined by criteria based on Bender's classification of groups with strongly 2-embedded subgroups.A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are monomorphisms between subgroups, and which satisfy certain axioms first formulated by Puig [Pg] and motivated in part by conjugacy relations among p-subgroups of a given finite group. A saturated fusion system is realizable if it is isomorphic to the fusion system defined by the conjugation relations within a Sylow p-subgroup of some finite group, and is exotic otherwise. One of our main goals is to try to understand when and how exotic fusion systems can occur, especially over 2-groups.A saturated fusion system F is reduced if O p (F ) = 1 and O p (F ) = O p ′ (F ) = F (see Definitions 1.1(c,e) and 1.9(a)). A saturated fusion system F is tame if it is realized by a group G such that the natural homomorphism from Out(G) to a certain group of outer automorphisms of F (more precisely, of an associated linking system) is split surjective (Definition 1.10). The main result in our earlier paper [AOV1] says roughly that exotic fusion systems can be detected via tameness of associated reduced fusion systems. More precisely, by [AOV1, Theorems A & B], if the "reduction" of a fusion system F is tame, then F is tame and hence realizable, while if a reduced fusion system is not tame, then it is the reduction of an exotic fusion system.A saturated fusion system is indecomposable if it does not split as a product of fusion systems over nontrivial p-groups. We can now state our main result.Theorem A. Let F be a reduced, indecomposable fusion system over a nontrivial 2-group of order at most 2 9 . Then F is the fusion system of a finite simple group, and is tame.Proof. This is shown in Theorems 4.1 (for 2-groups of order at most 64), 4.3 (order 2 7 ), 5.1 (order 2 8 ), and 6.1 (order 2 9 ).The next theorem follows from Theorem A and the above discussion.Theorem B. Each saturated fusion system over a 2-group of order at most 2 9 is realizable.