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11
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.
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.
Simple connectedness of the 3local geometry of the Monster
 J. Algebra
, 1997
"... We consider the 3local geometryM of the Monster group M introduced in [BF] as a locally dual polar space of the group Ω−8 (3) and independently in [RS] in the context of minimal plocal parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly in [DM] within th ..."
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We consider the 3local geometryM of the Monster group M introduced in [BF] as a locally dual polar space of the group Ω−8 (3) and independently in [RS] in the context of minimal plocal 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 Z3orbifold 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 inM which is the 3local geometry of the Fischer group M(24) [IS]. 1
GRAPE A Package for GAP by
"... 1.1 Installing the GRAPE Package... 5 1.2 Loading GRAPE........ 6 1.3 The structure of a graph in GRAPE. 7 ..."
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1.1 Installing the GRAPE Package... 5 1.2 Loading GRAPE........ 6 1.3 The structure of a graph in GRAPE. 7
PERMUTATION GROUPS GENERATED BY BINOMIALS
"... Abstract. Let G(q) be the group of permutations of F∗q generated by those permutations which can be represented as c 7 → acm + bcn with a, b ∈ F∗q and 0 < m < n < q. We show that there are infinitely many q for which G(q) is the group of all permutations of F∗q. This resolves a conjecture o ..."
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Abstract. Let G(q) be the group of permutations of F∗q generated by those permutations which can be represented as c 7 → acm + bcn with a, b ∈ F∗q and 0 < m < n < q. We show that there are infinitely many q for which G(q) is the group of all permutations of F∗q. This resolves a conjecture of Vasilyev and Rybalkin. 1.
A design and a geometry for the group Fi22
"... The Fischer group Fi22 acts as a rank 3 group of automorphisms of a symmetric 2(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorph ..."
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The Fischer group Fi22 acts as a rank 3 group of automorphisms of a symmetric 2(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it. In this paper we construct and investigate a partial linear space admitting the Fischer group Fi22 and defined combinatorially from a symmetric design also admitting this group. For details about the Fischer group we refer to the ATLAS of Finite Groups [3]. The Fischer group Fi22 has two conjugacy classes of subgroups of index 14080 isomorphic to O7(3). One of these subgroups has orbits of size 1, 3159, and 10920 on its own conjugacy class, and 364, 1080, and 12636 on the other. Construct an incidence structure in which the elements of the two classes are points and
A characterization of 3local geometry of M(24)
, 1996
"... The largest Fischer 3transposition group M(24) acts flagtransitively on a 3local incidence geometry G(M(24)) which is a cextension of the dual polar space associated with the group O 7 (3). The action of the simple commutator subgroup M(24) 0 is still flagtransitive. We show that G(M(24)) is ..."
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The largest Fischer 3transposition group M(24) acts flagtransitively on a 3local incidence geometry G(M(24)) which is a cextension of the dual polar space associated with the group O 7 (3). The action of the simple commutator subgroup M(24) 0 is still flagtransitive. We show that G(M(24)) is characterized by its diagram under the flagtransitivity assumption. The result implies in particular that G(M(24)) is simply connected. The geometry G(M(24)) appears as a subgeometry in the Buekenhout  Fischer 3local geometry G(F 1 ) of the Monster group. The simple connectedness of G(M(24)) has played a crucial role in the characterization of G(F 1 ), achieved recently. When determining the possible structure of the parabolic subgroups we have used an unpublished pushingup result by U. Meierfrankenfeld. 1 Introduction Let ffi be the following diagram: ffi : 1 ffi c 3 ffi 3 ffi 3 ffi: An (incidence) geometry G with diagram ffi has rank 4; the types of its elements are 1, 2, 3...