Abstract of article: 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.
About the author: Gary Gordon, a native Floridian, received his B.A. from the University of Florida in 1977 and his Ph.D. from the University of North Carolina in 1983. He has taught mathematics at Lafayette College since 1986, and before that, at Williams College. He has also worked in industry. His mathematical interests include combinatorics, geometry, and algebra. Most of his research has involved analyzing polynomial and other invariants for graphs, posets, and other combinatorial objects. He loves watching baseball and playing softball, tennis, golf, and other sports where people swing clubs. He also enjoys all sorts of games, but usually loses to his wife and frequent mathematical collaborator, Liz McMahon, and to his two daughters, Rebecca and Hannah.
Software mentioned: The Geometer's Sketchpad, Tesselmania, Kali, KaleidoTile