Motivated by the concept of symmetry, we present in this chapter the fundamental elements of group theory. Roughly speaking, symmetry is a transformation that leaves an object of study unchanged, meaning that the object looks the same from different points of view. The set of transformations that characterize the symmetry of an object naturally form a group. Group theory is a branch of mathematics that was inspired by these types of groups, and is of paramount importance in many areas of physics, chemistry, engineering, and computer science. In chemistry, for example, groups are used to classify crystal structures and the symmetries of molecules. In physics, groups are used for solving problems in atomic, molecular, and solid state physics. Groups are extensively used in cryptography, which is the science of encoding information so that only certain specified people can decode it. In addition, group theory has proven very useful in a wide variety of signal and image processing applications, including filter design, image edge detection, and deformable image registration and retrieval.

The outline of this chapter is as follows. In Section 2.1, we start by pointing out the interesting connection of groups with symmetry. Just as numbers can be used to measure size, groups can be used to measure symmetry. This relation of groups with symmetry reveals an important linkage between geometry and algebra. Then, we introduce the notion of a group, and describe in detail the group-theoretical concepts through illustrative examples. In particular, we take a detailed look at subgroups, cosets and normal subgroups, quotient groups, homomorphisms, cyclic and permutation groups. We also highlight matrix groups. Section 2.2 provides a very bare summary of some basic facts about topological spaces and metric spaces. We describe homeomorphisms between topological spaces and isometries between metric spaces, then highlight the connection between these two structure-preserving maps. Next, we introduce two important classes of groups, the topological and symmetry groups.