There is no (known) formula which gives the number of groups of order
n for any . However, it's possible to classify the
finite *abelian* groups of order n. This classification
follows from the * structure theorem for finitely
generated abelian groups*.

* Definition.* Let G be an abelian group. The * torsion subgroup* of G is

I'd better check that the definition makes sense!

* Proposition.* Let G be an abelian group. The
torsion subgroup of G is a subgroup of G.

* Proof.* Let T be the torsion subgroup of G.
, so T is nonempty. Let . I must show . Find
positive integers m, n, such that and . Then

Therefore, , and .

* Definition.* A group G is *
torsion free* if the only element of finite order is the
identity.

* Definition.* An abelian group G is * finitely generated* if there are elements such that every element can be written as

Note that this expression need not be unique.

* Definition.* A * free abelian
group* is a direct sum of copies of (possibly infinitely many copies).

The number of copies (in the sense of cardinality) is the * rank* of the free abelian group. It's possible to
prove that the rank of a free abelian group is well-defined.

* Theorem.* Let G be a finitely generated abelian
group.

(a) , where T is the torsion subgroup and F is a free abelian group.

(b) The rank of F is uniquely determined by G.

(c) The torsion part T can be written as a direct sum of cyclic groups in the following ways. Each decomposition is unique (in the first case, up to the order of the factors):

In the first case, the p's are primes (not necessarily distinct), and
for all i. The first case is called a * primary decomposition* while the second case is
called an * invariant factor decomposition*.

The proof of this result is outside the scope of this course. But I
should mention that it is related to the * Jordan
canonical form* and * rational canonical
form* that you may have seen in linear algebra. The structure
theorem for finitely generated abelian groups and the results on
canonical forms are special cases of a more general structure
theorem: The *structure theorem for finitely generated modules
over a principal ideal domain*.

Let's concentrate for now on the case of a *finite* abelian
group. Since any factor of would make the
group infinite, there can't be any 's in the
decomposition. The result then says that every finite abelian can be
written as

Here the p's are primes and the r's are positive integers (* primary decomposition*).

Alternatively, you can write the same group as

In this case, the d's are positive integers and (* invariant
factor decomposition*).

* Example.* (* Listing all the
primary and invariant factor decompositions*) Find the primary
decompositions and corresponding invariant factor decompositions for
all abelian groups of order 360.

First, factor 360 into a product of primes: .

Next, write each prime power in all possible ways:

You get the primary decompositions by using one of the factorizations, one of the factorizations, and the lone 5. I'll list the possibilities below, together with the corresponding invariant factor decompositions.

The two groups in each row are *isomorphic* --- they're
"the same" as groups.

Here's an example which shows how I got the invariant factor decompositions. Consider . Write the numbers for each prime in a row, right-justified:

Multiply the numbers in each column. These give the numbers for the invariant factor decomposition. Note that 2 divides 6 and 6 divides 30.

* Example.* (* Finding the
primary and invariant factor decompositions for a specific
group*) Find the primary decomposition and invariant factor
decomposition for .

First, I take each of the factors apart into direct products of groups of prime power order.

I'm using the fact that if and only if m and n are relatively prime. Thus, because 3 and 4 are relatively prime.

I can't replace with because 2 is not relatively prime to 2 (2 and 2 have the common factor 2!).

Thus, the primary decomposition is

Next, I find the invariant factor decomposition:

So the invariant factor decomposition is

Note that 2 divides 12 and 12 divides 36.

* Example.* (* Finding primary
decompositions satisfying a condition on orders of elements*)
Suppose G is an abelian group of order 24, and no element has order
greater than 12. What are the possible primary decompositions for G?

Since , the primary decompositions for abelian groups of order 24 are

Let . Then

Therefore, no element of has order greater than 12.

Let . Then

Therefore, no element of has order greater than 12.

However, for , I have

So does not have order less than 12 --- in fact, it has order 24.

Therefore, the possible primary decompositions for G are and .

Copyright 2018 by Bruce Ikenaga