Groups and Symmetry

1. Front Matter

   Pages i-xi

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2. Symmetries of the Tetrahedron

   • M. A. Armstrong

   Pages 1-5

3. Axioms

   • M. A. Armstrong

   Pages 6-10

4. Numbers

   • M. A. Armstrong

   Pages 11-14

5. Dihedral Groups

   • M. A. Armstrong

   Pages 15-19

6. Subgroups and Generators

   • M. A. Armstrong

   Pages 20-25

7. Permutations

   • M. A. Armstrong

   Pages 26-31

8. Isomorphisms

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   Pages 32-36

9. Plato’s Solids and Cayley’s Theorem

   • M. A. Armstrong

   Pages 37-43

10. Matrix Groups

    • M. A. Armstrong

    Pages 44-51

11. Products

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    Pages 52-56

12. Lagrange’s Theorem

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    Pages 57-60

13. Partitions

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    Pages 61-67

14. Cauchy’s Theorem

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    Pages 68-72

15. Conjugacy

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    Pages 73-78

16. Quotient Groups

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    Pages 79-85

17. Homomorphisms

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    Pages 86-90

18. Actions, Orbits, and Stabilizers

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    Pages 91-97

19. Counting Orbits

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    Pages 98-103

20. Finite Rotation Groups

    • M. A. Armstrong

    Pages 104-112

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

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

• Department of Mathematical Sciences, University of Durham, Durham, England

  M. A. Armstrong

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