Geometry of Sporadic Groups I : Petersen and Tilde Geometries (Encyclopedia of Math and its Applications)

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Geometry of Sporadic Groups I : Petersen and Tilde Geometries (Encyclopedia of Math and its Applications)

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  • 製本 Hardcover:ハードカバー版/ページ数 368 p.
  • 言語 ENG
  • 商品コード 9780521413626
  • DDC分類 512.2

Full Description


This book is the first volume in a two-volume set, which will provide the complete proof of classification of two important classes of geometries, closely related to each other: Petersen and tilde geometries. There is an infinite family of tilde geometries associated with non-split extensions of symplectic groups over a field of two elements. Besides that there are twelve exceptional Petersen and tilde geometries. These exceptional geometries are related to sporadic simple groups, including the famous Monster group and this volume gives a construction for each of the Petersen and tilde geometries which provides an independent existence proof for the corresponding automorphism group. Important applications of Petersen and Tilde geometries are considered, including the so-called Y-presentations for the Monster and related groups, and a complete indentification of Y-groups is given. This is an essential purchase for researchers into finite group theory, finite geometries and algebraic combinatorics.

Table of Contents

Preface                                            ix
1 Introduction 1 (48)
1.1 Basic definitions 2 (3)
1.2 Morphisms of geometries 5 (2)
1.3 Amalgams 7 (2)
1.4 Geometrical amalgams 9 (1)
1.5 Universal completions and covers 10 (1)
1.6 Tits geometries 11 (5)
1.7 Alt(7)-geometry 16 (1)
1.8 Symplectic geometries over GF(2) 17 (2)
1.9 From classical to sporadic geometries 19 (2)
1.10 The main results 21 (2)
1.11 Representations of geometries 23 (3)
1.12 The stages of classification 26 (7)
1.13 Consequences and development 33 (9)
1.14 Terminology and notation 42 (7)
2 Mathieu groups 49 (51)
2.1 The Golay code 50 (1)
2.2 Constructing a Golay code 51 (2)
2.3 The Steiner system S(5, 8, 24) 53 (3)
2.4 Linear groups 56 (3)
2.5 The quad of order (2, 2) 59 (3)
2.6 The rank 2 T-geometry 62 (2)
2.7 The projective plane of order 4 64 (7)
2.8 Uniqueness of S(5, 8, 24) 71 (3)
2.9 Large Mathieu groups 74 (2)
2.10 Some further subgroups of Mat(24) 76 (5)
2.11 Little Mathieu groups 81 (4)
2.12 Fixed points of a 3-element 85 (2)
2.13 Some odd order subgroups in Mat(24) 87 (3)
2.14 Involutions in Mat(24) 90 (5)
2.15 Golay code and Todd modules 95 (2)
2.16 The quad of order (3, 9) 97 (3)
3 Geometry of Mathieu groups 100(41)
3.1 Extensions of planes 101(1)
3.2 Maximal parabolic geometry of Mat(24) 102(4)
3.3 Minimal parabolic geometry of Mat(24) 106(6)
3.4 Petersen geometries of the Mathieu 112(5)
groups
3.5 The universal cover of G(Mat(22)) 117(5)
3.6 G(Mat(23)) is 2-simply connected 122(2)
3.7 Diagrams for H(Mat(24)) 124(6)
3.8 More on Golay code and Todd modules 130(2)
3.9 Diagrams for H(Mat(22)) 132(6)
3.10 Actions on the sextets 138(3)
4 Conway groups 141(69)
4.1 Lattices and codes 141(6)
4.2 Some automorphisms of lattices 147(3)
4.3 The uniqueness of the Leech lattice 150(3)
4.4 Coordinates for Leech vectors 153(5)
4.5 Co(1), Co(2) and Co(3) 158(2)
4.6 The action of Co(1) on Lambda(4) 160(3)
4.7 The Leech graph 163(6)
4.8 The centralizer of an involution 169(4)
4.9 Geometries of Co(1) and Co(2) 173(5)
4.10 The affine Leech graph 178(11)
4.11 The diagram of Delta 189(4)
4.12 The simple connectedness of G(Co(2)) 193(5)
and G(Co(1))
4.13 McL geometry 198(5)
4.14 Geometries of 3 . U(4)(3) 203(7)
5 The Monster 210(62)
5.1 Basic properties 211(5)
5.2 The tilde geometry of the Monster 216(2)
5.3 The maximal parabolic geometry 218(4)
5.4 Towards the Baby Monster 222(2)
5.5 2E(6)(2)-subgeometry 224(3)
5.6 Towards the Fischer group M(24) 227(4)
5.7 Identifying M(24) 231(5)
5.8 Fischer groups and their properties 236(6)
5.9 Geometry of the Held group 242(2)
5.10 The Baby Monster graph 244(12)
5.11 The simple connectedness of G(BM) 256(3)
5.12 The second Monster graph 259(6)
5.13 Uniqueness of the Monster amalgam 265(3)
5.14 On existence and uniqueness of the 268(3)
Monster
5.15 The simple connectedness of G(M) 271(1)
6 From C(n)- to T(n)-geometries 272(35)
6.1 On induced modules 273(3)
6.2 A characterization of G(3 . Sp(4)(2)) 276(4)
6.3 Dual polar graphs 280(5)
6.4 Embedding the symplectic amalgam 285(3)
6.5 Constructing T-geometries 288(2)
6.6 The rank 3 case 290(3)
6.7 Identification of J(n) 293(2)
6.8 A special class of subgroups in J(n) 295(2)
6.9 The I(n) are 2-simply connected 297(4)
6.10 A characterization of I(n) 301(2)
6.11 No tilde analogues of the 303(4)
Alt(7)-geometry
7 2-Covers of P-geometries 307(25)
7.1 On P-geometries 307(6)
7.2 A sufficient condition 313(2)
7.3 Non-split extensions 315(3)
7.4 G(3(23) . Co(2)) 318(3)
7.5 The rank 5 case: bounding the kernel 321(6)
7.6 G(3(4371) . BM) 327(3)
7.7 Some further s-coverings 330(2)
8 Y-groups 332(26)
8.1 Some history 333(2)
8.2 The 26-node theorem 335(2)
8.3 From Y-groups to Y-graphs 337(3)
8.4 Some orthogonal groups 340(5)
8.5 Fischer groups as Y-groups 345(6)
8.6 The monsters 351(7)
9 Locally projective graphs 358(40)
9.1 Groups acting on graphs 359(3)
9.2 Classical examples 362(5)
9.3 Locally projective lines 367(3)
9.4 Main types 370(4)
9.5 Geometrical subgraphs 374(5)
9.6 Further properties of geometrical 379(4)
subgraphs
9.7 The structure of P 383(3)
9.8 Complete families of geometrical 386(3)
subgraphs
9.9 Graphs of small girth 389(3)
9.10 Projective geometries 392(2)
9.11 Petersen geometries 394(4)
Bibliography 398(8)
Index 406