Saturated fusion systems over 2-groups

We develop methods for listing, for a given 2-group S, all nonconstrained centerfree saturated fusion systems over S. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order 2 and two of order 2. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order. For any prime p and any finite p-group S, a saturated fusion system over S is a category F whose objects are the subgroups of S, whose morphisms are injective group homomorphisms between the objects, and which satisfy certain axioms due to Puig and described here in Section 2. Among the motivating examples are the categories F = FS(G) where G is a finite group with Sylow p-subgroup S: the morphisms in FS(G) are the group homomorphisms between subgroups of S which are induced by conjugation by elements of G. A saturated fusion system F which does not arise in this fashion from a group is called “exotic”. When p is odd, it seems to be fairly easy to construct exotic fusion systems over p-groups (see, e.g., [BLO2, §9], [RV], and [Rz]), although we are still very far from having any systematic understanding of how they arise. But when p = 2, the only examples we know are those constructed by Levi and Oliver [LO], based on earlier work by Solomon [Sol] and Benson [Bs]. The smallest such example known is over a group of order 2, and it is possible that there are no exotic examples over smaller groups. Our goal in this paper is to take a first step towards developing techniques for systematically searching for exotic fusion systems, a search which eventually can be carried out in part using a computer. A fusion system F is constrained (Definition 2.3) if it contains a normal p-subgroup which contains its centralizer. Any constrained fusion system is the fusion system of a unique finite group with analogous properties [BCGLO1, Proposition C]. A fusion system F over S is centerfree (Definition 2.3) if there is no element 1 6= z ∈ Z(S) such that each morphism in F extends to a morphism between subgroups containing z which sends z to itself. By [BCGLO2, Corollary 6.14], if there is such a z, and if F is exotic, then there is a smaller exotic fusion system F/〈z〉 over S/〈z〉. Thus all minimal exotic fusion systems must be nonconstrained and centerfree, and these conditions provide a convenient class of fusion systems to search for and list. If F is a saturated fusion system over any p-group S, then the F-essential subgroups of S are the proper subgroups P S which “contribute new morphisms” to the category F : it is the smallest set of objects such that each morphism in S is a composite of restrictions of automorphisms of essential subgroups and of S itself. We 2000 Mathematics Subject Classification. Primary 20D20. Secondary 20D45, 20D08.