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A presentation for the Thompson sporadic simple group
, 2001
"... We determine a presentation for the Thompson sporadic simple group Th. The proof of correctness of this presentation uses a coset enumeration of 143,127,000 cosets. In the process of our work, we determine presentations for 3 D4 (2), 3 D4 (2):3, G2 (3):2, and CTh(2A) (of shape 2 1+8 + : A ..."
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We determine a presentation for the Thompson sporadic simple group Th. The proof of correctness of this presentation uses a coset enumeration of 143,127,000 cosets. In the process of our work, we determine presentations for 3 D4 (2), 3 D4 (2):3, G2 (3):2, and CTh(2A) (of shape 2 1+8 + : A9 ).
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
Simple connectedness of the 3local geometry of the Monster
 In: Moonshine, the Monster and Related Topics, C.Dong and G.Mason eds
, 1997
"... We consider the 3local geometry M of the Monster group M introduced in [BF] as a locally dual polar space of the group\Omega \Gamma 8 (3) and independently in [RS] in the context of minimal p local parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly i ..."
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We consider the 3local geometry M of the Monster group M introduced in [BF] as a locally dual polar space of the group\Omega \Gamma 8 (3) and independently in [RS] in the context of minimal p local parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly in [DM] within the Z3orbifold construction of the Moonshine module V " . In this paper we prove the simple connectedness of M. This result makes unnecessary the refereeing to the classification of finite simple groups in the Z3 orbifold construction of V " and realizes an important step in the classification of the flagtransitive cextensions of the classical dual polar spaces (cf. [Yo]). We make use of the simple connectedness results for the 2local geometry of M [Iv1] and for a subgeometry in M which is the 3local geometry of the Fischer group M(24) [IS]. 1 Introduction The Monster group M acts flagtransitively on a diagram geometry M which is described by the following diagram...
CONSTRUCTION OF FISCHER’S SPORADIC GROUP Fi ′ 24 INSIDE GL8671(13)
, 906
"... Abstract. In this article we construct an irreducible simple subgroup G = 〈q, y, t, w 〉 of GL8671(13) from an irreducible subgroup T of GL11(2) isomorphic to Mathieu’s simple group M24 by means of Algorithm 2.5 of [13]. We also use the first author’s similar construction of Fischer’s sporadic simple ..."
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Abstract. In this article we construct an irreducible simple subgroup G = 〈q, y, t, w 〉 of GL8671(13) from an irreducible subgroup T of GL11(2) isomorphic to Mathieu’s simple group M24 by means of Algorithm 2.5 of [13]. We also use the first author’s similar construction of Fischer’s sporadic simple group G1 = Fi23 described in [11]. He starts from an irreducible subgroup T1 of GL11(2) contained in T which is isomorphic to M23. In [7] J. Hall and L. S. Soicher published a nice presentation of Fischer’s original 3transposition group Fi24 [6]. It is used here to show that G is isomorphic to the simple commutator subgroup Fi ′ 24 of Fi24. We also determine a faithful permutation representation of G of degree 306936 with stabilizer G1 = 〈q, y, w 〉 ∼ = Fi23. It enabled MAGMA to calculate the character table of G automatically. Furthermore, we prove that G has two conjugacy classes of involutions z and u such that CG(u) = 〈q, y, t 〉 ∼ = 2Aut(Fi22). Moreover, we determine a presentation of H = CG(z) and a faithful permutation representation of degree 258048 for which we document a stabilizer. 1.
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