Groups and SymmetrySpringer Science & Business Media, 27 feb 1997 - 187 páginas Groups are important because they measure symmetry. This text, designed for undergraduate mathematics students, provides a gentle introduction to the vocabulary and many of the highlights of elementary group theory. Written in an informal style, the material is divided into short sections, each of which deals with an important result or a new idea. Throughout the book, emphasis is placed on concrete examples, often geometrical in nature, so that finite rotation groups and the 17 wallpaper groups are treated in detail alongside theoretical results such as Lagrange's theorem, the Sylow theorems, and the classification theorem for finitely generated abelian groups. A novel feature at this level is a proof of the Nielsen-Schreier theorem, using groups actions on trees. There are more than 300 exercises and approximately 60 illustrations to help develop the student's intuition. |
Índice
Symmetries of the Tetrahedron | 1 |
Axioms | 6 |
Numbers | 11 |
Dihedral Groups | 15 |
Subgroups and Generators | 20 |
Permutations | 26 |
Isomorphisms | 32 |
Platos Solids and Cayleys Theorem | 37 |
Actions Orbits and Stabilizers | 91 |
Counting Orbits | 98 |
Finite Rotation Groups | 104 |
The Sylow Theorems | 113 |
Finitely Generated Abelian Groups | 119 |
Row and Column Operations | 125 |
Automorphisms | 131 |
The Euclidean Group | 136 |
Matrix Groups | 44 |
Products | 52 |
Lagranges Theorem | 57 |
Partitions | 61 |
Cauchys Theorem | 68 |
Conjugacy | 73 |
Quotient Groups | 79 |
Homomorphisms | 86 |
Lattices and Point Groups | 145 |
Wallpaper Patterns | 155 |
Free Groups and Presentations | 166 |
Trees and the NielsenSchreier Theorem | 173 |
Bibliography | 181 |
183 | |
Otras ediciones - Ver todo
Términos y frases comunes
3-cycles abelian group automorphism axis belongs to H bijection c2mm centre CHAPTER commutator complex numbers conjugacy classes conjugate correspondence cube cyclic group cyclic permutations defined denote diagonal dihedral group E₂ edges element of G elements of order equivalence classes equivalence relation EXAMPLE Exercise Figure finite order form a group form a subgroup free group G is isomorphic G₁ G₂ gives glide reflection group G group of order H₁ hexagonal homomorphism identity element infinite integer inverse isometry isomorphic to Z2 Lagrange's theorem lattice Let G Let H linear m₁ Mathematics matrix n₁ non-zero normal subgroup orbit order of G ordered pairs orthogonal plane point group positive integer Proof quotient group r³s real numbers reduced word rotational symmetry group sends Show SO₂ subgroup of G subgroup of order subset Suppose Sylow torsion coefficients translation transpositions vector vertex vertices wallpaper group Z₂
Referencias a este libro
Human Symmetry Perception and Its Computational Analysis Christopher W. Tyler No hay ninguna vista previa disponible - 2002 |
Group Theoretical Methods and Their Applications E. Stiefel,A. Fässler No hay ninguna vista previa disponible - 1992 |