Symmetric Generation of Groups
With Applications to many of the Sporadic Finite Simple Groups
Seiten
2007
Cambridge University Press (Verlag)
978-0-521-85721-5 (ISBN)
Cambridge University Press (Verlag)
978-0-521-85721-5 (ISBN)
This comprehensive text develops the notion of symmetric generation from scratch and goes on to describe how the technique can be used to define and construct many of the sporadic simple groups (including the Mathieu groups, the Janko groups and the Higman-Sims group) in a uniform and accessible way.
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.
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.
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.
Preface; Acknowledgements; Part I. Motivation: 1. The Mathieu group M12; 2. The Mathieu group M24; Part II. Involutory Symmetric Generators: 3. The progenitor; 4. Classical examples; 5. Sporadic simple groups; Part III. Non-involutory Symmetric Generators: 6. The progenitor; 7. Images of these progenitors.
| Erscheint lt. Verlag | 5.7.2007 |
|---|---|
| Reihe/Serie | Encyclopedia of Mathematics and its Applications |
| Zusatzinfo | Worked examples or Exercises; 54 Tables, unspecified; 4 Plates, color; 49 Line drawings, unspecified; 4 Line drawings, color |
| Verlagsort | Cambridge |
| Sprache | englisch |
| Maße | 165 x 240 mm |
| Gewicht | 712 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| ISBN-10 | 0-521-85721-X / 052185721X |
| ISBN-13 | 978-0-521-85721-5 / 9780521857215 |
| Zustand | Neuware |
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