Groups, a path to geometry

This book follows the same successful approach as Dr Burn's previous book on number theory. It consists of a carefully constructed sequence of questions which will enable the reader, through his or her own participation, to generate all the group theory covered by a conventional first university course. An introduction to vector spaces, leading to the study of linear groups, and an introduction to complex numbers, leading to the study of Möbius transformations and stereographic projection, are also included. Quaternions and their relationship to three-dimensional isometries are covered, and the climax of the book is a study of crystallographic groups, with a complete analysis of these groups in two dimensions.

Cover -- Frontmatter -- Contents -- Preface -- Acknowledgements -- Functions -- Summary -- Historical note -- Answers -- Permutations of a finite set -- Summary -- Historical note -- Answers -- Groups of permutations of R and C -- Summary -- Historical note -- Answers -- The Möbius group -- Summary -- Historical note -- Answers -- The regular solids -- Summary -- Historical note -- Answers -- Abstract groups -- Summary -- Historical note -- Answers -- Inversions of the Möbius plane and stereographic projection -- Summary -- Historical note -- Answers -- Equivalence relations -- Summary -- Historical note -- Answers -- Cosets -- Summary -- Historical note -- Answers -- Direct product -- Summary -- Historical note -- Answers -- Fields and vector spaces -- Summary -- Historical note -- Answers -- Linear transformations -- Summary -- Historical note -- Answers -- The general linear group GL(2, F) -- Summary -- Historical note -- Answers -- The vector space V3(F) -- Summary -- Historical note -- Answers -- Eigenvectors and eigenvalues -- Summary -- Historical note -- Answers -- Homomorphisms -- Summary -- Historical note -- Answers -- Conjugacy -- Summary -- Historical note -- Answers …