Abstract. 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 7 and two of order 2 10. 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”.

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This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.

Abstract. In this article we give self-contained existence proofs for the sporadic simple groups Co2 and Fi22 using the second author’s algorithm [10] constructing finite simple groups from irreducible subgroups of GLn(2). These two sporadic groups were originally discovered by J. Conway [4] and B. Fischer [7], respectively, by means of completely different and unrelated methods. In this article n = 10 and the irreducible subgroups are the Mathieu group M22 and its automorphism group Aut(M22). We construct their five non-isomorphic extensions Ei by the two 10-dimensional non-isomorphic simple modules of M22 and by the two 10-dimensional simple modules of A22 = Aut(M22) over F = GF(2). In two cases we construct the centralizer Hi = CG i (zi) of a 2-central involution zi of Ei in any target simple group Gi. Then we prove that all the conditions of Algorithm 7.4.8 of [11] are satisfied. This allows us to construct G3 ∼ = Co2 inside GL23(13) and G2 ∼ = Fi22 inside GL78(13). We also calculate their character tables and presentations. 1.

Classification

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