Visual Group Theory : A Computer-Oriented Geometric Introduction

“Symmetry is everywhere. Flowers, crystals, many natural phenomena and artifacts appear beautiful to us, especially in their symmetrical appearance. One may enjoy this beauty and leave it at that in admiration. Here, a systematic approach to symmetry is proposed. Mathematically, it should be made as widely and as generally understandable as possible.In the narrower sense, this book is about group theory. Groups can be considered as algebraic objects that describe the symmetry of geometric objects. This will mainly be the perspective adopted in the following, and thus, this book is also a book about geometry. Groups describe symmetry phenomena algebraically—one calculates with reflections, rotations, etc., generally with mappings of spaces onto themselves.”Fragmento deVisual Group Theory 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.