Introduction to Groups
We now begin our study of abstract algebra in earnest! A group is one of the simplest algebraic structures; we take a set, assign an operation to it, impose four basic rules, and see what we can deduce. And yet, the possibilities are endless. Groups show up everywhere, and not just in mathematics. Indeed, it would be difficult to study physics or chemistry without an understanding of group theory. The solution to the famous Rubik’s cube is also a problem in groups. In this chapter, we will define the notion of a group, and give a number of examples. We will also prove several basic properties of groups and subgroups.