Direct Products of Groups
There are two sorts of products on groups that I’d like to discuss. Today I’ll talk about direct products.
The direct product says that we can take two groups, form the Cartesian product of their sets, and put the structure of a group on that. Given groups and we form the group as the set of pairs with in and in . We compose them term-by-term: . It can be verified that this gives us a group.
There’s a very interesting property about this group. It comes equipped with two homomorphisms, and , the “projections” of onto and , respectively. As one might expect, , and similarly for . Even better, let’s consider any other group with homomorphisms and . There is a unique homomorphism — defined by — so that and . Here’s the picture.
The vertical arrow from to is , and I assert that that’s the only homomorphism from to so that both paths from to are the same, as are both paths from to . When we draw a diagram like this with groups on the points and homomorphisms for arrows, we say that the diagram “commutes” if any two paths joining the same point give the same homomorphism between those two groups.
To restate it again, has homomorphisms to and , and any other group with a pair of homomorphisms to and has a unique homomorphism from to so that the above diagram commutes. This uniqueness means that has this property is unique up to isomorphism.
Let’s say two groups and have this product property. That is, each has given homomorphisms to and , and given any other group with a pair of homomorphisms there is a unique homomorphism to and one to that make the diagrams commute (with or in the place of ). Then from the diagram with in place of we get a unique homomorphism . On the other hand, from the diagram with in place of , we get a unique homomorphism . Putting these two together we get homomorphisms and .
Now if we think of the diagram for with itself in place of , we see that there’s a unique homomorphism from to itself making the diagram commute. We just made one called , but the identity homomorphism on also works, so they must be the same! Similarly, must be the identity on , so and are inverses of each other, and and are isomorphic!
So let’s look back at this whole thing again. I take two groups and , and I want a new group that has homomorphisms to and and so any other such group with two homomorphisms has a unique homomorphism to . Any two groups satisfying this property are isomorphic, so if we can find any group satisfying this property we know that any other one will be essentially the same. The group structure we define on the Cartesian product of the sets and satisfies just such a property, so we call it the direct product of the two groups.
This method of defining things is called a “universal property”. The argument I gave to show that the product is essentially unique works for any such definition, so things defined to satisfy universal properties are unique (up to isomorphism) if they actually exist at all. This is a viewpoint on group theory that often gets left out of basic treatments of the subject, but one that I feel gets right to the heart of why the theory behaves the way it does. We’ll definitely be seeing more of it.