Review of algebra

Abstract

In this chapter we discuss some of the theory of finite abelian groups and finitely generated abelian groups. Abelian groups form the indexing sets for data. This fact by itself is not sufficient for the central importance we place on this theory. The most important processing, representational, and algorithmic constructions and procedures in DSP depend ultimately on concepts that come from abelian group theory. The general theory is considered first. We introduce the groups most often occurring in DSP applications, Z/N, the group of integers mod N, direct products of such groups, and Z, the group of integers and its direct products. Subgroups, coset decompositions and quotient groups, homomorphisms and automorphisms are discussed and highlighted by several examples that are refined throughout the text.