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Symmetric Generation of Groups With Applications to Many of the Sporadic Finite Simple Groups

ISBN-10: 052185721X

ISBN-13: 9780521857215

Edition: 2007

Authors: Robert T. Curtis

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Some of the most beautiful mathematical objects found in the last forty years are the sporadic simple groups. But gaining familiarity with these groups presents problems for two reasons. Firstly, they were discovered in many different ways, so to understand their constructions in depth one needs to study lots of different techniques. Secondly, since each of them is in a sense recording some exceptional symmetry in spaces of certain dimensions, they are by their nature highly complicated objects with a rich underlying combinatorial structure. Motivated by initial results which showed that the Mathieu groups can be generated by highly symmetrical sets of elements, which themselves have a natural geometric definition, the author develops from scratch the notion of symmetric generation. He exploits this technique by using it to define and construct many of the sporadic simple groups including all the Janko groups and the Higman-Sims group. For researchers and postgraduates.
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Book details

List price: $199.99
Copyright year: 2007
Publisher: Cambridge University Press
Publication date: 7/5/2007
Binding: Hardcover
Pages: 332
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.562
Language: English

Robert T. Curtis is Professor of Combinatorial Algebra and Head of Pure Mathematics in the School of Mathematics at the University of Birmingham. He also holds the post of Librarian for the London Mathematical Society.

Introduction to Part I
The Mathieu group M[subscript 12]
The combinatorial approach
The regular dodecahedron
The algebraic approach
Independent proofs
The Mathieu group M[subscript 24]
The combinatorial approach
The Klein map
The algebraic approach
Independent proofs
Conclusions to Part I
Involutory Symmetric Generators
The (involutory) progenitor
Free products of cyclic groups of order 2
Semi-direct products and the progenitor P
The Cayley graph of P over N
The regular graph preserved by P
Homomorphic images of P
The lemma
Further properties of the progenitor
Coxeter diagrams and Y-diagrams
Introduction to Magma and GAP
Algorithm for double coset enumeration
Systematic approach
Classical examples
The group PGL[subscript 2](7)
Exceptional behaviour of S[subscript n]
The 11-point biplane and PGL[subscript 2](11)
The group of the 28 bitangents
Sporadic simple groups
The Mathieu group M[subscript 22]
The Janko group J[subscript 1]
The Higman-Sims group
The Hall-Janko group and the Suzuki chain
The Mathieu groups M[subscript 12] and M[subscript 24]
The Janko group J[subscript 3]
The Mathieu group M[subscript 24] as control subgroup
The Fischer groups
Transitive extensions and the O'Nan group
Symmetric representation of groups
Appendix to Chapter 5
Non-Involutory Symmetric Generators
The (non-involutory) progenitor
Monomial automorphisms
Monomial representations
Monomial action of a control subgroup
Images of the progenitors in Chapter 6
The Mathieu group M[subscript 11]
The Mathieu group M[subscript 23]
The Mathieu group M[subscript 24]
Factoring out a 'classical' relator
The Suzuki chain and the Conway group
Systematic approach
Tabulated results
Some sporadic groups