Visual Group Theory A Computer-Oriented Geometric Introduction

This textbook provides an introduction to group theory starting from the basics, relying on geometry to elucidate its various aspects.

Groups naturally manifest as symmetries of geometric shapes, such as reflections and rotations. The book adopts this perspective to provide a straightforward, descriptive explanation, supported by examples and exercises in GAP, an open-source computer algebra system. It covers all of the key concepts of group theory, including homomorphisms, group operations, presentations, products of groups, and finite, abelian, and solvable groups. The topics include cyclic and symmetric groups, dihedral, orthogonal, and hyperbolic groups, as well as the significant notion of Cayley graphs.

Self-contained and requiring little beyond high school mathematics, this book is aimed at undergraduate courses and features numerous exercises. It will also appeal to anyone interested in the geometric approach to group theory.

1 Introduction to Euclidean Geometry.- 2 Introduction to Groups.- 3 Subgroups and Homomorphisms.- 4 Group Operations.- 5 Group Presentations.- 6 Products of Groups.- 7 Finite Groups.- 8 Abelian and Solvable Groups.- 9 The Hyperbolic Plane.- 10 Hyperbolic Groups.

Stephan Rosebrock is a professor at the Institute of Mathematics and Computer Science of the University of Education in Karlsruhe, Germany.