Finitely generated abelian groups have a structure theorem, which makes them easy to describe and enumerate completely. The structure theorem states that every finitely generated abelian group is a direct sum of and where is a prime.

*Proof.* Every nontrivial subgroup of
is cyclic.
Any two nontrivial subgroups
have a non-trivial intersection at
. Thus
cannot be a
direct sum of those subgroups, hence
is
indecomposable. Suppose
is a nontrivial decomposition with
and
with
. Then
which is a contradiction since
has an
element of order
.Â â—»

In other words, the structure theorem says that every finitely generated abelian group is the direct sum of a finite number of indecomposable groups.

This lemma suggests that we can combine the prime powers for distinct primes for a different representation. Before we discuss the structure theorem, a helpful lemma:

Finally, we state the structure theorem.

We omit the proof as it is quite lengthy, but it only uses elementary methods. If is a finitely generated abelian group, then the uniquely determined integers are called the invariant factors of . The uniquely determined prime powers are called the elementary divisors of .

As an example, consider describing all finite abelian groups of order 1500 up to isomorphism. . The only possible families of elementary divisors are In general, the number of families of elementary divisors depends on the integer partitions of the powers of the primes in the prime decomposition. Each of these six families determines an abelian group of order 1500. E.g. determines . From the list of families of elementary divisors, we can derive an equivalent list of families of invariant factors, and vice versa. Suppose that an invariant factor decomposition were known. Then the elementary divisors are the prime powers that arise from the prime factorizations of . Conversely, if the elementary divisors are known, one can arrange them in a matrix: where are distinct primes, with some and for some . While this is a sufficient description, some observations that make this process easier to visualize: the last row contains the highest prime powers for each prime with no zero exponents. The first row contains nonzero exponents for the primes with the most amount of prime power terms in the family. Let . By construction, and we have constructed the invariant factors.

As an example, consider . Then, The elementary divisors of are thus . This can be arranged as Thus the invariant factors of are so that .

Here is a simple exercise from the section that I liked: prove that a finite abelian group that is not cyclic contains a subgroup which is isomorphic to for some prime .

Let be a finite abelian group that is not cyclic. Consider its invariant factor decomposition: . If , then would be cyclic hence . Since and , and must share a prime factor . Then . Take the subgroup corresponding to .

In the next post, we discuss the Krull-Schmidt theorem which extends this notion of uniquely decomposing a group into a finite direct product of indecomposable subgroups to groups that satisfy both the ascending chain condition or descending chain condition on normal subgroups.