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Groups /
This text provides an introduction to group theory with an emphasis on clear examples. The authors present groups as naturally occurring structures arising from symmetry in geometrical figures and other mathematical objects. Written in a 'user-friendly' style, where new ideas are always mo...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Oxford :
Newnes,
1994.
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| Colección: | Modular mathematics series.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Front Cover; Groups; Copyright Page; Series Preface; Preface; Table of Contents; Chapter 1. Squares and Circles; 1.1 Symmetries of a square; 1.2 Symmetries of a circle; 1.3 Further exercises on Chapter 1; Chapter 2. Permutations; 2.1 The symmetric group S4; 2.2 Functions; 2.3 Permutations; 2.4 Basic properties of cycles; 2.5 Cycle decomposition; 2.6 Transpositions; 2.7 The 15-puzzle; 2.8 Further exercises on Chapter 2; Chapter 3. Linear Transformations and Matrices; 3.1 Matrix multiplication; 3.2 Linear transformations; 3.3 Orthogonal matrices; 3.4 Further exercises on Chapter 3
- Chapter 4. The Group Axioms4.1 Number systems; 4.2 Binary operations; 4.3 Definition of a group; 4.4 Examples of groups; 4.5 Consequences of the axioms; 4.6 Direct products; 4.7 Further exercises on Chapter 4; Chapter 5. Subgroups; 5.1 Subgroups; 5.2 Examples of subgroups; 5.3 Groups of symmetries; 5.4 Further exercises on Chapter 5; Chapter 6. Cyclic Groups; 6.1 Cyclic groups; 6.2 Cyclic subgroups; 6.3 Order of elements; 6.4 Orders of products; 6.5 Orders of powers; 6.6 Subgroups of cyclic groups; 6.7 Direct products of cyclic groups; 6.8 Further exercises on Chapter 6
- Chapter 7. Group Actions7.1 Groups acting on sets; 7.2 Orbits; 7.3 Stabilizers; 7.4 Permutations arising from group actions; 7.5 The alternating group; 7.6 Further exercises on Chapter 7; Chapter 8. Equivalence Relations and Modular Arithmetic; 8.1 Partitions; 8.2 Relations; 8.3 Equivalence classes; 8.4 Equivalence relations from group actions; 8.5 Modular arithmetic; 8.6 Further exercises on Chapter 8; Chapter 9. Homomorphisms and Isomorphisms; 9.1 Comparing D3 and S3; 9.2 Properties of homomorphisms; 9.3 Homomorphisms arising from group actions; 9.4 Cayley's theorem; 9.5 Cyclic groups
- 9.6 Further exercises on Chapter 9Chapter 10. Cosets and Lagrange's Theorem; 10.1 Left cosets; 10.2 Left cosets as equivalence classes; 10.3 Lagrange's theorem; 10.4 Consequences of Lagrange's theorem; 10.5 Applications to number theory; 10.6 Right cosets; 10.7 Further exercises on Chapter 10; Chapter 11. The Orbit-Stabilizer Theorem; 11.1 The orbit-stabilizer theorem; 11.2 Fixed subsets; 11.3 Counting orbits; 11.4 Further exercises on Chapter 11; Chapter 12. Colouring Problems; 12.1 Colouring problems; 12.2 Groups of symmetries in three dimensions; 12.3 Three-dimensional colouring problems
- 12.4 Further exercises on Chapter 12Chapter 13. Conjugates, Centralizers and Centres; 13.1 Conjugates; 13.2 Conjugacy classes; 13.3 Conjugacy classes in Sn; 13.4 Centralizers; 13.5 Centres; 13.6 Conjugates and centralizers; 13.7 Further exercises on Chapter 13; Chapter 14. Towards Classification; 14.1 An action of S3 on three-dimensional space; 14.2 Cauchy's theorem; 14.3 Direct products; 14.4 Further exercises on Chapter 14; Chapter 15. Kernels and Normal Subgroups; 15.1 Kernels of homomorphisms; 15.2 Kernels of actions; 15.3 Conjugates of a subgroup; 15.4 Normal subgroups