Geometric groups.

Abstract

Our objective in this paper is to discuss and illustrate basic properties of geometric groups and some of their applications in chemistry. In Chapter 1, we provide the readers with basic concepts from linear algebra and abstract algebra that are needed in later chapters. In Chapter 2, we study two types of length (or distance) preserving transformations of a finite-dimensional Euclidean space, namely, orthogonal transformations and Euclidean transformations. In Chapter 3, we state and prove Cartan's Theorem and apply it to the classification of orthogonal and Euclidean transformations on 2-and 3-dimensional Euclidean spaces. In Chapter 4, we define the symmetry group of a set in a Euclidean space and classify the finite symmetry groups of bounded sets in the 2-and 3-dimensional Euclidean spaces, R2 and R3. In Chapter 5, we present two applications of geometric groups namely, the study of the trigonometric functions of a 2-dimensional Euclidean space and isomer enumeration in organic chemistry via P6lya's Theorem.