ABSTRACT

When mathematicians study a structure like groups, they often take two different approaches. One is to find a representation for the group. What this typically means is to define an (often) injective homomorphism from the group into a larger structure that provides us with a more concrete notation and terminology, and with better computational techniques. In Chapter 17 we discovered that groups of symmetries of geometric figures could be better understood by representing them either as subgroups of a symmetric group of permutations, or else as a subgroup of a larger group of matrices. In linear algebra we learn a lot about computing with matrices, and in Chapter 18 we learned a good bit about computing with permutations. In this chapter we will discover that all (finite) groups can be represented as a subgroup of a symmetric group of permutations. (It is also actually true that any finite group can be represented as a group of matrices, but we will avoid the details in this book, because such representations often require matrices of large size, and more about linear algebra than we wish to use.)

An alternative to a representation is to look at the internal structure of the group. Can we better understand how a group works by looking at its subgroups and how they are related? The Chinese Remainder Theorem is a spectacular example of how this approach works for rings (and hence abelian groups) of the form Zn. In this chapter we will examine how groups can sometimes be thought of as a direct product of smaller pieces (or subgroups). This chapter will provide only a small start on this approach, and we will explore more sophisticated approaches in Chapters 28, 29, and 30.