The Unapologetic Mathematician

Mathematics for the interested outsider

A difficult exercise

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 A_1, A_2, B_1, and B_2, and four homomorphisms:

  • f_{1,1}:A_1\rightarrow B_1
  • f_{1,2}:A_1\rightarrow B_2
  • f_{2,1}:A_2\rightarrow B_1
  • f_{2,2}:A_2\rightarrow B_2

Use these to construct two homomorphisms from A_1*A_2 to B_1\times B_2, and show that they’re the same.

March 1, 2007 Posted by | Algebra, Group Examples, Group theory, Universal Properties | 2 Comments

More no news

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.

March 1, 2007 Posted by | Uncategorized | 2 Comments

Free products of groups

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.

Universal Property of Coproducts

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 G and H, the free product is a group G*H with homomorphisms \iota_G:G\rightarrow G*H and \iota_H:H\rightarrow G*H so that given any other group X with homomorphisms f_G:G\rightarrow X and f_H:H\rightarrow X there is a unique homomorphism f_G*f_H:G*H\rightarrow X.

The exact same argument from Tuesday shows that if there is any such group G*H, it is uniquely determined up to isomorphism. What we need is to have an example of a group satisfying this property.

First of all, \iota_G has to be an monomorphism. Just put G itself for X, the identity on G for f_G, and the trivial homomorphism (sending everything to the identity) for f_H. Now anything in the kernel of \iota_G is automatically in the kernel of the composition of \iota_G with the coproduct map. But this composite is the identity homomorphism on G, which has trivial kernel, and so must \iota_G. This means that a copy of G must sit inside G*H. Similarly a copy of H must sit inside G*H.

So how do those two copies interact? Let’s put F_2 in for X and let f_G send everything in G to some power of the generator a — effectively defining a homomorphism from G to F_1\cong\mathbb{Z} — while f_H sends everything in H to some power of the generator b. If there’s any relation between the copies of G and H sitting inside G*H, it won’t be respected by the function to F_2 required by the universal property. So there can’t be any such relation.

G*H is actually very much like F_2, only instead of alternating powers of a and b, we have alternating members of G and H. An arbitrary element looks something like g_1h_1g_2h_2...g_kh_k. Of course it could start with an element of H or end with an element of G. 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 G and the next sequence starts with an element of G, we compose those elements in G so the whole sequence is still alternating.

Does G sit inside here? Of course! It’s just sequences with only an element of G in them. The same goes for H. And given any group X and homomorphisms f_G and f_H, we can send g_1h_1g_2h_2...g_k to f_G(g_1)f_H(h_1)f_G(g_2)f_H(h_2)...f_G(g_k). That’s our f_G*f_H. As I said above, any other group that has this property is isomorphic to G*H, so we’re done.

If we compare the free product G*H with the direct product G\times H, we see that the main difference is that elements of G and H don’t commute inside G*H, but they do inside G\times H. We can check that (g,e_H)(e_G,h)=(ge_G,e_Hh)=(g,h)=(e_Gg,he_H)=(e_G,h)(g,e_H). In fact, take a presentation of G with generators X and relations R, and one of H with generators Y and relations S, and with X and Y sharing no elements. Then G*H has generators X\cup Y and relations R\cup S, while G\times H has generators X\cup Y and relations R\cup S\cup\{xyx^{-1}y^{-1} (x\in X, y\in Y)\}.

Since we’ve only added some relations to the presentation to get from G*H to G\times H, the latter group is a quotient of the former. There should be some epimorphism from G*H to G\times H. I’ll leave it to you to show that some such epimorphism does exist in two ways: once by the universal property of G*H and once by the universal property of G\times H.

March 1, 2007 Posted by | Algebra, Group theory, Structure of Groups, Universal Properties | 6 Comments

God vs. The Bible

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.

March 1, 2007 Posted by | Uncategorized | 2 Comments

An article in the February Notices

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.

March 1, 2007 Posted by | Uncategorized | Leave a comment

   

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