This book contains a first course in group theory, pursued with conventional rigour. There are three unusual aspects of the presentation.

Firstly, the book consists of a sequence of over 800 problems. This is to enable the course to proceed by seminar rather than by lecture. Mathematics is something we do rather than something we learn, and, all too often, lectures give the opposite impression.

Secondly, at the outset, the groups under discussion are groups of transformations. This is faithful to the historical origins of the theory. It provides the one context in which the proof of the associative law is immediate, and it makes the study of sets with only a single defined operation obviously worthwhile. For Galois (1830), Jordan (1870) and even in Klein's ‘Lectures on the Icosahedron’ (1884), groups were defined by the one axiom of closure. The other axioms were implicit in the context of their discussions – finite groups of transformations. Our work on abstract groups starts in chapter 6.

Thirdly, the geometry of two and three dimensions is the context in which most of the groups in this book are constructed, and is also the major field of application of group theory in chapters 7, and 17–23. Geometry is the best context in which to understand conjugacy, and linear and affine groups are some of the easiest in which to put homomorphisms to work to good effect.