# Gracious Living

The Classification Theorem for Finitely Generated Abelian Groups
January 29, 2011, 16:23
Filed under: Algebra, Math | Tags: , , , ,

Wow, it’s been a long time since I’ve written anything on this blog.  I’m taking algebraic topology and an algebraic number theory course this semester, and I started reading through Atiyah and MacDonald’s Commutative Algebra over the winter.  So I thought I’d continue with a little algebra.  The algebra we’ve done thus far has been highly noncommutative, for the most part — we investigated groups like free groups, symmetric groups, matrix groups, and dihedral groups in which the order of operations mattered.  As you might expect, with abelian groups, the theory becomes much simpler, and the subject called “commutative algebra” is just the study of abelian groups with extra structure — something like a scalar multiplication, as in the case of vector spaces, or some other operation.  But first, we need to understand abelian groups.

When talking about abelian groups specifically, we usually write them additively: the group operation applied to $a$ and $b$ is $a+b$, and then we can build expressions like $3a+2b$.  The proof I give below is due to J. S. Milne, who in turn says it’s similar to Kronecker’s original proof.  Of course, I’ve added more detail in places where I thought it was necessary, and taken it out where I thought it wasn’t.  There are other, more common proofs, typically using matrices, but I find them unwieldy and inelegant.