Edited by Martin W. Liebeck
Edited by Jan Saxl
Publisher: Cambridge University Press
Print Publication Year: 1992
Online Publication Date:September 2010
Online ISBN:9780511629259
Paperback ISBN:9780521406857
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511629259.003
Subjects: Algebra, Discrete Mathematics, Information Theory and Coding
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Initial work on the sporadic finite simple groups falls into one or more of the following categories:
Discovery
Structure
Existence
Uniqueness
More precisely let H be some group theoretic hypothesis. A group theorist begins to investigate groups G satisfying H and generates information about the structure of such groups. Typical examples of structural information include the group order, the isomorphism type of normalizers of subgroups of prime order, and perhaps eventually the character table of G. When a sufficiently large body of self-consistent structural information has been generated, the group is said to be discovered. This is roughly the point where the group theoretic community first becomes convinced that the group exists.
The group actually exists when there is a proof that there is at least one group satisfying hypothesis H, while the group is unique when there is a proof that, up to isomorphism, there is at most one group satisfying H. More detailed information about the group structure usually comes later and might include the calculation of the automorphism group and Schur multiplier of G, an enumeration of the maximal subgroups of G, and the generation of the modular character tables for G.
As part of the ongoing effort to produce a complete, unified, and accessible proof of the Classification Theorem, Aschbacher has begun to try to write down in one place a complete and fairly self-contained proof that the 26 sporadic groups exist and are unique. The plan is to generate at the same time the basic structural information about each sporadic group necessary for the Classification.
pp. i-iv
pp. v-vii
pp. viii-x
pp. xi-xiv
Part 1 - Sporadic groups
1 - Uniqueness of sporadic groups: Read PDF
pp. 1-11
2 - The study of J4 via the theory of uniqueness systems: Read PDF
pp. 12-21
3 - Y555 and all that: Read PDF
pp. 22-23
4 - Hyperbolic reflections for the Bimonster and 3Fi24 : Read PDF
pp. 24-45
5 - A geometric characterization of the Monster: Read PDF
pp. 46-62
6 - Constructing the Monster: Read PDF
pp. 63-76
Part 2 - Moonshine
7 - The signature of the normalizes of Γ0 (N): Read PDF
pp. 77-86
8 - Completely replicahle functions: Read PDF
pp. 87-98
9 - Introduction to the Monster Lie algebra: Read PDF
pp. 99-107
10 - Remarks on Moonshine and Orbifolds: Read PDF
pp. 108-120
Part 3 - Local and geometric methods in group theory
11 - The classification of 3-transposition groups with trivial center: Read PDF
pp. 121-138
12 - (S3,S6,)-Amalgams: Read PDF
pp. 139-143
13 - Pushing down minimal parabolic systems: Read PDF
pp. 144-150
14 - Nonspherical spheres: Read PDF
pp. 151-158
15 - On the 2-local structure of finite groups: Read PDF
pp. 159-182
16 - Groups generated by k-root subgroups – a survey: Read PDF
pp. 183-204
Part 4 - Geometries and related groups
17 - Finiteness questions for geometries: Read PDF
pp. 205-217
18 - Kac-Moody groups and, their automorphisms: Read PDF
pp. 218-228
19 - Generalized hexagons as geometric hyperplanes of near hexagons: Read PDF
pp. 229-239
20 - On simplicial complexes related to the Suzuki sequence graphs: Read PDF
pp. 240-248
21 - Twin buildings and groups of Kac-Moody type: Read PDF
pp. 249-286
Part 5 - Finite and algebraic groups of Lie type
22 - Some remarks on the structure of finite subgroups of simple algebraic groups: Read PDF
pp. 287-291
23 - Some (almost) multiplicity-free coset actions: Read PDF
pp. 292-310
24 - Orbits in internal Chevalley modules: Read PDF
pp. 311-315
25 - Subgroups of finite and algebraic groups: Read PDF
pp. 316-326
26 - Irreducible representations of finite Chevalley groups containing a matrix with a simple spectrum: Read PDF
pp. 327-332
27 - Overgroups of unipotent elements in simple algebraic groups: Read PDF
pp. 333-339
Part 6 - Finite permutation groups
28 - Some open problems on permutation groups: Read PDF
pp. 340-350
29 - The genus of a permutation group: Read PDF
pp. 351-363
30 - Primitive permutation characters: Read PDF
pp. 364-367
31 - Closures of finite permutation groups and relation algebras: Read PDF
pp. 368-379
Part 7 - Further aspects of simple groups
32 - Symmetric presentations I: Introduction, with particular reference to the Mathicu groups M12 and M24 : Read PDF
pp. 380-396
33 - Finite and, locally finite groups containing a small subgroup with small centralizer: Read PDF
pp. 397-402
34 - Some topics in asymptotic group theory: Read PDF
pp. 403-421
35 - The 3-modular characters of the McLaughlin group McL and its automorphism group McL.2: Read PDF
pp. 422-437
Part 8 - Related topics
36 - The orbifold notation for surface groups: Read PDF
pp. 438-447
37 - A remark on two Diophantine equations of Peter Cameron: Read PDF
pp. 448-458
38 - Testing for isomorphism between finitely presented groups: Read PDF
pp. 459-475
39 - Discrete groups and, Galois theory: Read PDF
pp. 476-479
40 - Smooth coverings of regular maps: Read PDF
pp. 480-489