UID:
almafu_9959245973202883
Format:
1 online resource (xiii, 408 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88190-6
,
1-107-38370-6
,
1-107-39493-7
,
1-299-90915-9
,
0-511-95398-4
,
0-511-83312-1
,
1-107-39014-1
,
1-107-39856-8
,
0-511-52593-1
Series Statement:
Encyclopedia of mathematics and its application ; v. 76
Content:
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.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Half-title; Title; Copyright; Contents; Preface; 1 Introduction; 1.1 Basic definitions; 1.2 Morphisms of geometries; 1.3 Amalgams; 1.4 Geometrical amalgams; 1.5 Universal completions and covers; 1.6 Tits geometries; 1.7 Alt7-geometry; 1.8 Symplectic geometries over GF(2); 1.9 From classical to sporadic geometries; 1.10 The main results; 1.11 Representations of geometries; 1.12 The stages of classification; 1.13 Consequences and development; 1.14 Terminology and notation; 2 Mathieu groups; 2.1 The Golay code; 2.2 Constructing a Golay code; 2.3 The Steiner system S(5,8,24)
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2.4 Linear groups2.5 The quad of order (2,2); 2.6 The rank 2 T-geometry; 2.7 The projective plane of order 4; 2.8 Uniqueness of S(5,8,24); 2.9 Large Mathieu groups; 2.10 Some further subgroups of Mat24; 2.11 Little Mathieu groups; 2.12 Fixed points of a 3-element; 2.13 Some odd order subgroups in Mat24; 2.14 Involutions in Mat24; 2.15 Golay code and Todd modules; 2.16 The quad of order (3,9); 3 Geometry of Mathieu groups; 3.1 Extensions of planes; 3.2 Maximal parabolic geometry of Mat24; 3.3 Minimal parabolic geometry of Mat24; 3.4 Petersen geometries of the Mathieu groups
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3.5 The universal cover of g(Mat22)3.6 g(Mat23) is 2-simply connected; 3.7 Diagrams for H(Mat24); 3.8 More on Golay code and Todd modules; 3.9 Diagrams for H(Mat22); 3.10 Actions on the sextets; 4 Conway groups; 4.1 Lattices and codes; 4.2 Some automorphisms of lattices; 4.3 The uniqueness of the Leech lattice; 4.4 Coordinates for Leech vectors; 4.5 Co1, C02 and C03; 4.6 The action of Co1 on Ā4; 4.7 The Leech graph; 4.8 The centralizer of an involution; 4.9 Geometries of Co1and C02; 4.10 The affine Leech graph; 4.11 The diagram of; 4.12 The simple connectedness of g{Co2) and g{Co\)
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4.13 McL geometry4.14 Geometries of 3 U4(3); 5 The Monster; 5.1 Basic properties; 5.2 The tilde geometry of the Monster; 5.3 The maximal parabolic geometry; 5.4 Towards the Baby Monster; 5.5 2E6(2)-subgeometry; 5.6 Towards the Fischer group M(24); 5.7 Identifying M(24); 5.8 Fischer groups and their properties; 5.9 Geometry of the Held group; 5.10 The Baby Monster graph; 5.11 The simple connectedness of g(BM); 5.12 The second Monster graph; 5.13 Uniqueness of the Monster amalgam; 5.14 On existence and uniqueness of the Monster; 5.15 The simple connectedness of g(M)
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6 From Cn- to Tn-geometries6.1 On induced modules; 6.2 A characterization of g(3 Sp4(2)); 6.3 Dual polar graphs; 6.4 Embedding the symplectic amalgam; 6.5 Constructing T -geometries; 6.6 The rank 3 case; 6.7 Identification of J(n); 6.8 A special class of subgroups in J(n); 6.9 The f(n) are 2-simply connected; 6.10 A characterization of f(n); 6.11 No tilde analogues of the Alt7-geometry; 7 2-Covers of P -geometries; 7.1 On P -geometries; 7.2 A sufficient condition; 7.3 Non-split extensions; 7.4 g(323 Co2); 7.5 The rank 5 case: bounding the kernel; 7.6 g(34371 BM)
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7.7 Some further s-coverings
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English
Additional Edition:
ISBN 0-521-06283-7
Additional Edition:
ISBN 0-521-41362-1
Language:
English
