Using Geometry in Teaching Group Theory

Discrete groups of isometries provide a rich class of groups for detailed study in an undergraduate course in abstract algebra. After developing some geometric tools, we illustrate several important ideas from group theory via symmetry groups. In particular, we consider conjugation, homomorphisms, normal and nonnormal subgroups, generators and relations, quotient groups and direct and semidirect products. Understanding of these topics can be enhanced using symmetries of frieze and wallpaper patterns. We describe several examples in detail and indicate how other geometric investigations could be undertaken. We also indicate how four commercial software packages (The Geometer’s Sketchpad, KaleidoTile, TesselMania and Kali) can be used to enhance students’ understanding.