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19
SATURATED FUSION SYSTEMS OVER 2GROUPS
"... Abstract. We develop methods for listing, for a given 2group 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 tes ..."
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Abstract. We develop methods for listing, for a given 2group 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 2groups of small order. For any prime p and any finite pgroup 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 psubgroup 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”.
Simultaneous Constructions of the Sporadic Groups Co2
, 906
"... Abstract. In this article we give selfcontained 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. ..."
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Abstract. In this article we give selfcontained 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 nonisomorphic extensions Ei by the two 10dimensional nonisomorphic simple modules of M22 and by the two 10dimensional simple modules of A22 = Aut(M22) over F = GF(2). In two cases we construct the centralizer Hi = CG i (zi) of a 2central 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.
The maximal 2local subgroups of the Monster and Baby Monster, in preparation
, 2002
"... The lists of the maximal 2local subgroups of the Monster and Baby Monster simple groups in the Atlas are complete. 1 ..."
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The lists of the maximal 2local subgroups of the Monster and Baby Monster simple groups in the Atlas are complete. 1
Coding theory and algebraic combinatorics
, 2008
"... 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 part ..."
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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.
STRONGLY CLOSED SUBGROUPS OF FINITE GROUPS
, 809
"... Abstract. This paper gives a complete classification of the finite groups that contain a strongly closed psubgroup for p any prime. 1. ..."
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Abstract. This paper gives a complete classification of the finite groups that contain a strongly closed psubgroup for p any prime. 1.
Notes on the local theory of saturated fusion systems. Unpublished lecture notes for the Univ. of Birmingham short course on fusion systems
 31 July—4
, 2007
"... These notes are intended to supply an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems ” I mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite pgroups. One can ask: Why deal with ..."
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These notes are intended to supply an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems ” I mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite pgroups. One can ask: Why deal with saturated fusion systems rather than plocal finite groups? There are two reasons for this choice. First, as far as I know, it is not yet known whether to each saturated fusion system there is associated a unique plocal finite group. Thus it remains possible that the class of saturated fusion systems is larger than the class of plocal finite groups. But more important, to date there is no accepted notion of a morphism of plocal finite groups, and hence no category of plocal groups. The local theory of finite groups is inextricably tied to the notion of group homomorphism and factor group, so to extend the local theory of finite groups to a different category, we must at the least be dealing with an actual category. The first four sections of these notes record various basic definitions, notation, and notions from the theory of saturated fusion systems. Most of this material is taken from [BLO], and some of it was first written down by Puig. In addition in section 4 we record the deeper result of [BCGLO1] that if F is saturated and constrained on S, then the set G(F) of models of F is nonempty. Here G ∈ G(F) if G is a finite group with S ∈ Sylp(G), CG(Op(G)) ≤ Op(G), and F = FS(G). This fact is the basis for much of the local theory of fusion systems, and allows us to translate suitable statements from the local theory of groups into the setting of fusion systems. In Exercise 2.4, we see that if α: F → ˜ F is a morphism of fusion systems, then the kernel ker(α) of the group homomorphism α: S → ˜ S is strongly closed in S with respect to F. In section 5, we see how to construct a factor system F/T of F over a
United States of America, "Plaintiff v
 USA
, 1992
"... One of the best ways to understand the nature of finite simple groups is through geometries associated with them. This approach for classical and exceptional groups of Lie type has been quite successful and has led to the deveopment of the concept of buildings and polar spaces. The latter have been ..."
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One of the best ways to understand the nature of finite simple groups is through geometries associated with them. This approach for classical and exceptional groups of Lie type has been quite successful and has led to the deveopment of the concept of buildings and polar spaces. The latter have been characterized by simple systems of axioms with a combinatorialgeometric flavour. It has been observed recently that geometries similar to buildings can be associated with finite sporadic simple groups. However, most of the known characterizations of such geometries for sporadic groups require additional assumptions of a grouptheoretic nature. One aim of this thesis is to present characterizations of geometries for the sporadic groups J2, Suz, McL, Co3, Fi22, Fi23, Fi24 and He, which are in the same spirit as the characterizations of buildings and polar spaces mentioned above, in particular without any assumption on the automorphism groups of the geometries. A byproduct of these results for J2, Suz and He is a proof that certain presentations for those groups are faithful. Most of this work may be viewed as a contribution to the theory of graphs
computed. It follows that G and Fi23 have the same character table. REPRESENTATION THEORETIC EXISTENCE PROOF FOR FISCHER GROUP Fi23 3
, 2008
"... In the first section of this senior thesis the author provides some new efficient algorithms for calculating with finite permutation groups. They cannot be found in the computer algebra system Magma, but they can be implemented there. For any finite group G with a given set of generators, the algori ..."
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In the first section of this senior thesis the author provides some new efficient algorithms for calculating with finite permutation groups. They cannot be found in the computer algebra system Magma, but they can be implemented there. For any finite group G with a given set of generators, the algorithms calculate generators of a fixed subgroup of G as short words in terms of original generators. Another new algorithm provides such a short word for a given element of G. These algorithms are very useful for documentation and performing demanding experiments in computational group theory. In the later sections, the author gives a selfcontained existence proof for Fischer’s sporadic simple group Fi23 of order 2 18 · 3 13 · 5 2 · 7 · 11 · 13 · 17 · 23 using G. Michler’s Algorithm [11] constructing finite simple groups from irreducible subgroups of GLn(2). This sporadic group was originally discovered by B. Fischer in [6] by investigating 3transposition groups, see also [5]. This thesis gives a representation theoretic and algorithmic existence proof for his group. The author constructs
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
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