The Structure Theorem for Finitely Generated Abelian Groups

This paper provides a thorough explication of the Structure Theorem for Abelian groups and of the background information necessary to prove it. The outline of this paper is as follows. We first consider some theorems related to abelian groups and to R-modules. In this section we see that every finitely generated abelian group is the epimorphic image of a finitely generated free abelian group. Here also submodules of a finitely generated module over a principal ideal domain are shown to be finitely generated as well. Notably, this proposition is proved by examining a short exact sequence of R-modules. Then, in the next section, we learn a procedure for diagonalizing m×n matrices with products of elementary matrices. Moreover, these products can be interpreted as change-of-basis matrices. We will see the relevance of this procedure in the proof of the Stacked Basis Theorem. This theorem shows that for a subgroup of a finitely generated group, there is a basis of that subgroup whose elements are multiples of the basis elements of the group. Finally, the propositions enumerated in these sections allow us to prove and understand the Structure Theorem for Abelian Groups. This theorem asserts that every finitely generated abelian group is the direct sum of cyclic groups and of a free abelian group. Throughout this paper, examples and definitions will be included to aid the reader in understanding the relevant concepts.