Filed under: Algebra, Math | Tags: abelian groups, algebra, arithmetic, field theory, group theory, Math, ring theory
In which I sort of breeze through a couple of really awesome and really important concepts. Last time, we classified abelian groups — now we’ll see what happens if we require additional structure on the groups. In particular, I’m going to construct and similarly to how the Peano axioms constructed .
Filed under: Algebra, Math | Tags: abelian groups, algebra, commutative, group theory, Math
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 and is , and then we can build expressions like . 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.
Filed under: Algebra, Math | Tags: abelian groups, algebra, categorical, group theory, Math
Before looking at solvability and group classification, I want to mention a couple more ways of “building” groups. We’ve already seen how to find subgroups, and how to take the quotient by a normal subgroup, and how to find the direct product of a family of groups. Dual to the direct product is the free product, which generalizes the idea of a free group. The amalgamated free product is just a free product that we neutralize on the image of some map. Also, though the only really good example is the group of Euclidean isometries, the semidirect product is worth a more formal look. Finally, though it’s mostly terminology, I define the direct sum, which is useful for studying abelian groups.