The last way to induce a topological space is by taking a quotient. This can be compared to taking a quotient of groups in the same way that the product topology corresponds to a direct product of groups, the subspace topology corresponds to a subgroup, and so on. The quotient topology is a bit less intuitive than the other constructions we’ve done, but once you get the hang of it, it turns out to be very useful.

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.

Filed under: Algebra, Math | Tags: algebra, categorical, group theory, MaBloWriMo, Math

Ugh, so, I’ve been really busy today and haven’t had the time to do a Banach-Tarski post. Since I really do want to see MaBloWriMo to the end, I’m going to take a break from the main exposition and quickly introduce something useful. There are a couple major ways of combining two groups into one. The most important one, called the direct product, is analogous to the product of topological spaces. I know this is sort of a wussy post — sorry.

Filed under: Math, Topology | Tags: categorical, MaBloWriMo, Math, pretty pictures, topology

As the subspace topology is the “best way” to topologize a subset, the product and disjoint union topologies are the “natural ways” to topologize Cartesian products and disjoint unions of topological spaces. Usually the disjoint union topology is only glossed over, while more time is spent on the product topology; I’m introducing them together in order to show you some of the similarities between them addressed by category theory. The relationship is one of *duality*, something like the intersection and union of sets. There’s this goofy mathematician way of putting “co” in front of dual constructions, so we could also call the disjoint union the “coproduct”.