© 1988

Groups and Symmetry


Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. M. A. Armstrong
    Pages 1-5
  3. M. A. Armstrong
    Pages 6-10
  4. M. A. Armstrong
    Pages 11-14
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    Pages 15-19
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    Pages 20-25
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    Pages 26-31
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  20. M. A. Armstrong
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About this book


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.


Abelian group Group theory Lattice Point group automorphism group action

Authors and affiliations

  1. 1.Department of Mathematical SciencesUniversity of DurhamDurhamEngland

Bibliographic information

  • Book Title Groups and Symmetry
  • Authors Mark A. Armstrong
  • Series Title Undergraduate Texts in Mathematics
  • DOI
  • Copyright Information Springer-Verlag New York 1988
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-387-96675-5
  • Softcover ISBN 978-1-4419-3085-9
  • eBook ISBN 978-1-4757-4034-9
  • Series ISSN 0172-6056
  • Edition Number 1
  • Number of Pages XI, 187
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Group Theory and Generalizations
  • Buy this book on publisher's site


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