# Finitely Generated Abelian Groups

Finitely generated abelian groups arise all over algebraic number theory. For example, they will appear in this book as class groups, unit groups, and the underlying additive groups of rings of integers, and as Mordell-Weil groups of elliptic curves.

In this section, we prove the structure theorem for finitely generated abelian groups, since it will be crucial for much of what we will do later.

Let denote the ring of (rational) integers, and for each positive integer  let denote the ring of integers modulo , which is a cyclic abelian group of order  under addition.

Definition 2.1.1 (Finitely Generated)   A group is finitely generated if there exists such that every element of can be expressed as a finite product of positive or negative powers of the .

For example, the group is finitely generated, since it is generated by .

Theorem 2.1.2 (Structure Theorem for Abelian Groups)   Let be a finitely generated abelian group. Then there is an isomorphism

where and . Furthermore, the and  are uniquely determined by .

We will prove the theorem as follows. We first remark that any subgroup of a finitely generated free abelian group is finitely generated. Then we see that finitely generated abelian groups can be presented as quotients of finite rank free abelian groups, and such a presentation can be reinterpreted in terms of matrices over the integers. Next we describe how to use row and column operations over the integers to show that every matrix over the integers is equivalent to one in a canonical diagonal form, called the Smith normal form. We obtain a proof of the theorem by reinterpreting in terms of groups. Finally, we observe by a simple argument that the representation in the theorem is necessarily unique.

Proposition 2.1.3   If is a subgroup of a finitely generated abelian group then is finitely generated.

The key reason that this is true is that  is a finitely generated module over the principal ideal domain . We will give a complete proof of a beautiful generalization of Proposition 2.1.3 in the context of Noetherian rings in Section 2.2, but will not prove this proposition here.

Corollary 2.1.4   Suppose is a finitely generated abelian group. Then there are finitely generated free abelian groups and and a homomorphism such that .

Proof. Let be generators for . Let and let be the map that sends the th generator of to . Then is a surjective homomorphism, and by Proposition 2.1.3 the kernel of is a finitely generated abelian group. Let and fix a surjective homomorphism . Then is isomorphic to .

Suppose is a nonzero finitely generated abelian group. By the corollary, there are free abelian groups and and a homomorphism such that . Choosing a basis for and , we obtain isomorphisms and for integers and . We can thus view as being given by left multiplication by the matrix whose columns are the images of the generators of in . The cokernel of this homomorphism is the quotient of by the image of (the -span of the columns of ), and this cokernel is isomorphic to .

By augmenting with zero columns or adding (standard basis) rows to , we may assume that . For example, we would replace

by

and would replace

by

The following proposition implies that we may choose a bases for and such that the matrix of is diagonal, so that the structure of the cokernel of will be easy to understand.

Proposition 2.1.5 (Smith normal form)   Suppose  is an integer matrix. Then there exist invertible integer matrices and such that is a diagonal matrix with entries , where , and . Here and are invertible as integer matrices, so and are . The matrix is called the Smith normal form of .

We will see in the proof of Theorem 2.1.2 that is uniquely determined by . An example of a matrix in Smith normal form is

Proof. The matrix will be a product of matrices that define elementary row operations and will be a product corresponding to elementary column operations. The elementary row and column operations over are as follows:
1. [Add multiple] Add an integer multiple of one row to another (or a multiple of one column to another).
2. [Swap] Interchange two rows or two columns.
3. [Rescale] Multiply a row by .
Each of these operations is given by left or right multiplying by an invertible matrix  with integer entries, where  is the result of applying the given operation to the identity matrix, and  is invertible because each operation can be reversed using another row or column operation over the integers.

To see that the proposition must be true, assume and perform the following steps (compare [Art91, pg. 459]):

1. By permuting rows and columns, move a nonzero entry of with smallest absolute value to the upper left corner of . Now attempt to make all other entries in the first row and column 0 by adding multiples of row or column 1 to other rows (see step 2 below). If an operation produces a nonzero entry in the matrix with absolute value smaller than , start the process over by permuting rows and columns to move that entry to the upper left corner of . Since the integers are a decreasing sequence of positive integers, we will not have to move an entry to the upper left corner infinitely often.

2. Suppose is a nonzero entry in the first column, with . Using the division algorithm, write , with . Now add times the first row to the th row. If , then go to step 1 (so that an entry with absolute value at most is the upper left corner). Since we will only perform step 1 finitely many times, we may assume . Repeating this procedure we set all entries in the first column (except ) to 0. A similar process using column operations sets each entry in the first row (except ) to 0.

3. We may now assume that is the only nonzero entry in the first row and column. If some entry of is not divisible by , add the column of containing to the first column, thus producing an entry in the first column that is nonzero. When we perform step 2, the remainder will be greater than 0. Permuting rows and columns results in a smaller . Since can only shrink finitely many times, eventually we will get to a point where every is divisible by . If is negative, multiple the first row by .
After performing the above operations, the first row and column of are zero except for which is positive and divides all other entries of . We repeat the above steps for the matrix obtained from by deleting the first row and column. The upper left entry of the resulting matrix will be divisible by , since every entry of is. Repeating the argument inductively proves the proposition.

Example 2.1.6   The matrix has Smith normal form to , and the matrix has Smith normal form As a double check, note that the determinants of a matrix and its Smith normal form match, up to sign. This is because

We compute each of the above Smith forms using SAGE, along with the corresponding transformation matrices. Warning: Currently in Sage the entries down the diagonal are reversed from the discussion above. First the matrix.

sage: A = matrix(ZZ, 2, [-1,2, -3,4])
sage: S, U, V = A.smith_form(); S
[2 0]
[0 1]
sage: U*A*V
[2 0]
[0 1]
sage: U
[ 1 -1]
[ 0  1]
sage: V
[4 1]
[3 1]

The SAGE matrix command takes as input the base ring, the number of rows, and the entries. Next we compute with a matrix.
sage: A = matrix(ZZ, 3, [1,4,9,  16,25,36, 49,64,81])
sage: S, U, V = A.smith_form(); S
[72  0  0]
[ 0  3  0]
[ 0  0  1]
sage: U*A*V
[72  0  0]
[ 0  3  0]
[ 0  0  1]
sage: U
[  1 -20 -17]
[  0   1  -1]
[  0   0   1]
sage: V
[  93   74   47]
[-156 -125  -79]
[  67   54   34]


Finally we compute the Smith form of a matrix of rank :

sage: m = matrix(ZZ, 3, [2..10]); m
[ 2  3  4]
[ 5  6  7]
[ 8  9 10]
sage: m.smith_form()[0]
[0 0 0]
[0 3 0]
[0 0 1]


Proof. [Theorem 2.1.2] Suppose is a finitely generated abelian group, which we may assume is nonzero. As in the paragraph before Proposition 2.1.5, we use Corollary 2.1.4 to write as a the cokernel of an integer matrix . By Proposition 2.1.5 there are isomorphisms and such that is a diagonal matrix with entries , where and . Then is isomorphic to the cokernel of the diagonal matrix , so

 (2.1)

as claimed. The are determined by , because is the smallest positive integer  such that requires at most generators. We see from the representation (2.1.1) of as a product that has this property and that no smaller positive integer does.

William Stein 2012-09-24