Introduction A saturated fusion system F over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are injective homomorphisms between subgroups, and where the morphism sets satisfy certain axioms first formulated by Puig and motivated by the properties of conjugacy relations among p-subgroups of a finite group. In particular, for each finite group G and each Sylow p-subgroup S ≤ G, the category FS(G) whose objects are the subgroups of G and whose morphisms are those homomorphisms induced by conjugation in G is a saturated fusion system over S. We refer to Puig’s paper [Pg], and to [AKO] and [Cr], for more background details on saturated fusion systems. A saturated fusion system F is reduced if it contains no nontrivial normal p- subgroups, and no proper normal subsystems of p-power index or of index prime to p. All of these concepts are defined by analogy with finite groups the precise definitions are given in Section 1.2. The class of reduced fusion systems is larger than that of simple fusion systems, although a reduced fusion system which is not simple has to be fairly large. We refer to main theorems in [AOV1] for the motivation for defining this class. The sectional p-rank of a finite group G is the largest possible value of rk(P/Q), where Q P ≤ G are p-subgroups and P/Q is elementary abelian. When G is a p-group, we just call this the sectional rank, and denote it r(G). In their book which appeared in 1974, Gorenstein and Harada [GH] gave a classification of all finite simple groups whose sectional 2-rank is at most 4. A fusion system is indecomposable if it is not isomorphic to a product of fusion systems over smaller p-groups. The following theorem, where we list all reduced, indecomposable fusion systems over finite 2-groups of sectional rank at most 4, is the main result of this paper. We refer to the end of the introduction for the notation used for certain central products and semidirect products. When q is a prime power and n ≥ 2, UT n (q) denotes the group of upper triangular matrices over Fq with 1’s on the diagonal. Also, we write L+(q) n = PSLn(q) and L−(q) n = PSU n (q). A fusion system is simple if it contains no nontrivial proper normal subsystems. We refer to [AKO, Definition I.6.1] for the precise definition of a normal subsystem. Here, we just note that a reduced fusion system F over S is simple if S contains no nontrivial proper subgroup strongly closed in F. Theorem A. Let F be a reduced, indecomposable fusion system over a non- trivial 2-group S of sectional rank at most 4. Then one of the following holds. (1) S ∼ D2k for some k ≥ 3, and F is isomorphic to the fusion system of L2 + (q) (when v2(q2 − 1) = k + 1). (2) S ∼ SD2k for some k ≥ 4, and F is isomorphic to the fusion system of L3 ± (q) (when v2(q ± 1) = k − 2). 1

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