Groups and Symmetry

✍ Scribed by Mark Anthony Armstrong

• Publisher: Springer
• Year: 1988
• Tongue: English
• Leaves: 197
• Series: Undergraduate texts in mathematics
• Edition: 1
• Category: Library

✦ Synopsis

Groups are important because they measure symmetry. This text, designed for undergraduate mathematics students, provides a gentle introduction to 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, the emphasis is placed on concrete examples, many of them geometrical in nature, so that finite rotation groups and the seventeen 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 group actions on trees. There are more than three hundred exercises and approximately sixty illustrations to help develop the student's intuition.

✦ Table of Contents

Front Matter....Pages i-xi
Symmetries of the Tetrahedron....Pages 1-5
Axioms....Pages 6-10
Numbers....Pages 11-14
Dihedral Groups....Pages 15-19
Subgroups and Generators....Pages 20-25
Permutations....Pages 26-31
Isomorphisms....Pages 32-36
Plato’s Solids and Cayley’s Theorem....Pages 37-43
Matrix Groups....Pages 44-51
Products....Pages 52-56
Lagrange’s Theorem....Pages 57-60
Partitions....Pages 61-67
Cauchy’s Theorem....Pages 68-72
Conjugacy....Pages 73-78
Quotient Groups....Pages 79-85
Homomorphisms....Pages 86-90
Actions, Orbits, and Stabilizers....Pages 91-97
Counting Orbits....Pages 98-103
Finite Rotation Groups....Pages 104-112
The Sylow Theorems....Pages 113-118
Finitely Generated Abelian Groups....Pages 119-124
Row and Column Operations....Pages 125-130
Automorphisms....Pages 131-135
The Euclidean Group....Pages 136-144
Lattices and Point Groups....Pages 145-154
Wallpaper Patterns....Pages 155-165
Free Groups and Presentations....Pages 166-172
Trees and the Nielsen-Schreier Theorem....Pages 173-180
Back Matter....Pages 181-187

✦ Subjects

Group Theory and Generalizations