Groups and Symmetry

by Mark A. Armstrong

Overview

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.

Product Details

ISBN-13:	9781441930859
Publisher:	Springer New York
Publication date:	12/01/2010
Series:	Undergraduate Texts in Mathematics , #1
Edition description:	Softcover reprint of hardcover 1st ed. 1988
Pages:	187
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1 Symmetries of the Tetrahedron.- 2 Axioms.- 3 Numbers.- 4 Dihedral Groups.- 5 Subgroups and Generators.- 6 Permutations.- 7 Isomorphisms.- 8 Plato’s Solids and Cayley’s Theorem.- 10 Products.- 11 Lagrange’s Theorem.- 12 Partitions.- 13 Cauchy’s Theorem.- 14 Conjugacy.- 15 Quotient Groups.- 16 Homomorphisms.- 17 Actions, Orbits, and Stabilizers.- 18 Counting Orbits.- 19 Groups.- 20 The Sylow Theorems.- 21 Finitely Generated Abelian Groups.- 22 Row and Column Operations.- 23 Automorphisms.- 24 The Euclidean Group.- 25 Lattices and Point Groups.- 26 Wallpaper Patterns.- 27 Free Groups and Presentations.- 28 Trees and the Nielsen-Schreier Theorem.

Editorial Reviews

M.A. Armstrong

Groups and Symmetry

"This book is a gentle introductory text on group theory and its application to the measurement of symmetry. It covers most of the material that one might expect to see in an undergraduate course . . . The theory is amplified, exemplified and properly related to what this part of algebra is really for by discussion of a wide variety of geometrical phenomena in which groups measure symmetry. Overall, the author’s plan, to base his treatment on the premise that groups and symmetry go together, is a very good one, and the book deserves to succeed."—MATHEMATICAL REVIEWS

From the Publisher

Textbook introduces the vocabulary and some highlights of elementary group theory. Emphasis is placed on concrete examples so that finite rotation groups and the 17 wallpaper groups are treated in detail alongside theoretical results. Over 300 exercises and some 60 illustrations. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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