While proctoring the make-up exam for my class, I thought of an exercise related to my group theory posts on direct and free products that should cause even the more experienced mathematicians in the audience a bit of difficulty.
Consider four groups , , , and , and four homomorphisms:
Use these to construct two homomorphisms from to , and show that they’re the same.
The University of Colorado at Boulder was nice enough to point out in their rejection that they had narrowed the field of interest, and it wasn’t mine.
Also, Yale likes to tease me. There was a big envelope in my mailbox today that got my hopes up, but it turned out only to contain the university’s financial report. With a couple dozen pages about how well-endowed they are, it’s a sure thing that someone’s getting screwed.
On Tuesday I talked about how to put a group structure on the Cartesian product of two groups, and showed that it satisfied a universal property. For free products I’m just going to start from the property. First, the picture.
This is the same diagram as before, but now it’s upside-down. All the arrows run the other way. So we read this property as follows: Given two groups and , the free product is a group with homomorphisms and so that given any other group with homomorphisms and there is a unique homomorphism .
The exact same argument from Tuesday shows that if there is any such group , it is uniquely determined up to isomorphism. What we need is to have an example of a group satisfying this property.
First of all, has to be an monomorphism. Just put itself for , the identity on for , and the trivial homomorphism (sending everything to the identity) for . Now anything in the kernel of is automatically in the kernel of the composition of with the coproduct map. But this composite is the identity homomorphism on , which has trivial kernel, and so must . This means that a copy of must sit inside . Similarly a copy of must sit inside .
So how do those two copies interact? Let’s put in for and let send everything in to some power of the generator — effectively defining a homomorphism from to — while sends everything in to some power of the generator . If there’s any relation between the copies of and sitting inside , it won’t be respected by the function to required by the universal property. So there can’t be any such relation.
is actually very much like , only instead of alternating powers of and , we have alternating members of and . An arbitrary element looks something like . Of course it could start with an element of or end with an element of . The important thing is that the entries from the two groups alternate. We compose by just sticking sequences together like for a free group. If one sequence ends with an element of and the next sequence starts with an element of , we compose those elements in so the whole sequence is still alternating.
Does sit inside here? Of course! It’s just sequences with only an element of in them. The same goes for . And given any group and homomorphisms and , we can send to . That’s our . As I said above, any other group that has this property is isomorphic to , so we’re done.
If we compare the free product with the direct product , we see that the main difference is that elements of and don’t commute inside , but they do inside . We can check that . In fact, take a presentation of with generators and relations , and one of with generators and relations , and with and sharing no elements. Then has generators and relations , while has generators and relations .
Since we’ve only added some relations to the presentation to get from to , the latter group is a quotient of the former. There should be some epimorphism from to . I’ll leave it to you to show that some such epimorphism does exist in two ways: once by the universal property of and once by the universal property of .
According to WordPress, someone found this weblog yesterday by Googling “God vs. the Bible by John Armstrong” (no quotes in the search). It seemed an odd search, and predictably it turned up my “Gnosticism vs. Empiricism” post. But what could it have been intended to find?
Well here it is: God vs. The Bible, by John Armstrong. Mr. Armstrong describes himself as a deist, and interestingly his subject matter is not unrelated to my own thoughts. In the author’s own words, “The natural universe, God’s Creation, speaks out loudly and clearly against the Bible and other vaunted books of so-called ‘revelation’.” Further, “By the end of this book, the reader will have an undeniable case that humans, not God, wrote the Bible and that the only Word of God is the Creation itself.” Faith in the literal truth of a purported revelation stacked up against observations of the Created world. Sounds like gnosticism and empiricism to me.
Walt, over at Ars Mathematica points out that the February issue of the Notices of the AMS is up online. I won’t go into it all, but I did want to point out the “What Is…” column on Young Diagrams. I’ll be getting to these myself somewhere down the road, since they’re deeply related to conjugacy classes in symmetric groups.