## Contents of "Groups"

1. Squares and Circles.

2. Permutations.

3. Linear Transformations and Matrices.

4. The Group Axioms.

5. Subgroups.

6. Cyclic Groups.

7. Group Actions.

8. Equivalence Relations and Modular Arithmetic.

9. Homomorphisms and Isomorphisms.

10. Cosets and Lagrange's Theorem.

11. The Orbit-Stabilizer Theorem.

12. Colouring Problems.

13. Conjugates, Centralizers and Centres.

14. Towards Classification.

15. Kernels and Normal Subgroups.

16. Factor Groups.

17. Groups of Small Order.

18. Past and Future.

## Preface of "Groups"

Groups are sets of objects, which might be numbers, matrices or functions, in which objects can be combined, subject to a list of basic rules or axioms. The general theory is developed logically from these axioms. Although this idea of axiomatization is important, it can make the subject appear unduly abstract. Therefore we have made every effort, throughout the book, to illustrate new ideas through concrete examples.

Groups arise most naturally in considerations of symmetry. In particular, the symmetry of a geometrical figure, like a square or circle, can be measured by its group of symmetries. These groups are particularly helpful in illustrating many basic notions. The first chapter informally discusses these groups and introduces many ideas which are formalized later in the book.

We introduce the idea of a group acting on a set earlier than is perhaps usual. This involves thinking of elements of groups as doing something to elements of another set rather than as things satisfying a seemingly arbitrary list of axioms. For example, in the group of symmetries of the square, the elements of the group rotate or reflect the points of the square.

There are numerous exercises in the book. Those which are contained within chapters are usually short. Some are simply intended to help in understanding a new concept. Others encourage investigation of examples and, occasionally, prediction of results to be proved later. Solutions to these exercises are provided at the end of the book.

Each chapter ends with a section of further exercises. Some of these contain the word ``investigate". as an indication that the exercise is intended to be open ended. Many of these exercises would be suitable for tutorial discussion or workshop projects. Some suggestions for further projects, investigations and reading are included in the last chapter. This chapter also contains some historical notes on the development of the subject. These notes include brief comments on several mathematicians whose names appear in bold type elsewhere in the book.

The book is intended for use on a twenty-four lecture modular course. As there is variety in the timing of the introduction of group theory in different degree courses, and consequently in the background knowledge of students, we have included more material than we would expect to cover in twenty-four lectures. The first twelve chapters provide one possible course, stopping short of factor groups. The book could also be used by anyone who wishes to teach group theory as far as factor groups to students already familiar with general ideas such as functions and equivalence relations.

There are many people to whom thanks are due. Over the years we have both benefited from the cooperation and ideas of our colleagues in Sheffield. We would particularly like to thank John Greenlees, who made several suggestions, and the series editor Johnston Anderson, who made a number of comments. On a more practical note we should thank Mike Piff who helped us with LaTeX, in which we typed the original manuscript, and with the mfpic macros which we used for the pictures. We should also thank our sons Jonathan and Thomas who have tolerated many a mealtime discussion of ``the book". Finally we would like to thank the staff of Arnolds, both past and present, for their encouragement and help.

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