Abstract mathematical groups are everywhere in mathematics. A central, fundamental, mathematical abstraction, groups arise irresistibly in number theory, linear algebra, differential equations, algebraic topology, and differential geometry. Where there are symmetries, transformations, and invariants, there are mathematical groups. Where there is measurement, there are groups. Whenever one changes one’s choice of measurement units or system of coordinates, a mathematical group lurks in the background. An axis of measurement is an identified symmetry.

Groups are abstract, their study is difficult, and their key concepts require more motivation than they generally receive. Within the curriculum, groups arise, at least implicitly in treatments of linear transformations and matrices, but a more systematic study is for math majors and the important and fascinating study of group representations is a specialized topic for graduate students.

But, given an understanding of Chapter 7, the central concepts and motivation of group theory can be understood and appreciated without such specialized detail study. And so the purpose of Chapter 8 is to answer such questions as:

• How do mathematical groups relate to measurement?
• How do group-theoretic concepts relate to the world?
• What kinds of questions have driven the development of group theory?
• What motivates the transition from transformation groups to abstract groups?

Chapter 8 starts with a very simple concrete example and uses it to develop basic concepts of group theory and group representations, showing how such concepts arise even in the simplest of examples. Continuing the example as a central reference point, the chapter ends with a conceptual introduction to group representations that explains its importance and key concepts.