Wednesday, September 26, 2012

Green versus U of W: The gloves are off

Back in January, I told my readers that I had been expelled from the teacher certification program at the University of Winnipeg. Over the next two months, I attempted to fight my expulsion. The University set up a kangaroo court to hear my final appeal, which was rejected in the end.

Last week, I filed a Statement of Claim at the Manitoba Court of Queen's Bench claiming damages from the University of Winnipeg and various individuals who were involved in removing me from the program. I have just learned that the existence of my lawsuit is likely to be announced in the media very soon. So the fight goes public.

People in my situation have tried to sue many universities in the past, and the courts are not in general sympathetic to us. The university will undoubtedly attempt to have my claim dismissed outright before even going to trial, and they might succeed. Anything can happen when you go before a judge. On the other hand, I know what they did to me, and they know what they did, and I can tell you that they don't want to have to defend their actions in either the media or in court.

While I was still fighting my expulsion through the internal process, I was posting information on my other blogsite, "Due Process, Natural Justice and the University of Winnipeg". At that time, I didn't want to give away anything to the university that might compromise my position in any subsequent litigation, so I was only posting information that they already knew. At some point, I decided that it might not be in my best interest to have that information public, so I removed my posts. Tomorrow the fight becomes public, so I have restored all my old posts. To get the whole story, you have to start at the very oldest post and click through them one by one.

And that's where we stand as of today. I think I have a pretty good case against the university, but the judicial process can drag on for years, and even a huge settlement at the end can never make right what was taken away from me. I was a good teacher. I only spent a few weeks in the classroom but I knew that was where I belonged, and the kids knew it too. But my name has been blackened by the University and their collaborators, and now I have virtually no chance of getting back into the teaching profession.

I will be keeping my readers up to date on the fight as it develops from now on. You can follow my exploits on my "Due Process..." blog. In the meantime, I need to support myself for the duration of the fight, and I am therefore accepting private students. I have created a teaching blogsite called "Math with Marty" where I have started posting free on-line lesson modules for the high school math and science curriculum. If you like the on-line lessons, you will love the live in-person version.


Monday, September 17, 2012

What is a quotient group?

Group theory is all about permutations. The elements of a group are the various operations whereby you can re-arrange a set of elements. I used the word "elements" twice in that sentence, and I did it on purpose. It's confusing because the word "elements" refers to two different things: you have the "elements" A,B,C and D which we are going to shuffle around, and you have the operations such as "flips" and "rotation" which are the "elements" of the permutation group on the four elements (letters).

There are obvious way of getting around this confusion, but mathematicians have a way of being just a little too clever about these things. They do something which seems brilliant but turns out to be nasty in the end. They figured out that it's unnecessary to talk about the "elements" A,B,C, and D. Everything you need to know about the structure and properties of a group can be worked out by constructing the algebra of flips and rotations (F's and R's). You don't need to think about what is being flipped or rotated. I wrote about this yesterday.

It's all very elegant but it leads to a very awkward situation when you come to deal with the topic of quotient groups. Quotient Groups are something that makes a lot of sense in terms of permutations of elements like A,B,C and D. Let's consider the set of all permutations on these four letters; as we remarked yesterday, there are 24 possible re-shufflings (including the trivial one which leaves the letters as they were). What gets interesting is when we look at the set of functions of A,B,C, and D defined as:

O1 = AB + CD
O2 = AC + BD
O3 = AD + BC

I talked about these three functions last spring when I wrote about solving the fourth degree equation. The interesting thing about them is what happens when you re-shuffle the four letters. We've already noted that there are 24 possible re-shufflings of the letters A,B,C, and D. But you ought to be able to convince yourself that no matter what you do, the result of the re-shufflings is going to simply return the three functions you started with. They might be re-arranged: you might flip O1 and O2, or you might rotate O1 => O2 => O3. Or the thetas might just stay where they were. The overall effect is that from the twenty-four possible permutations of the four elements A,B,C and D, you have generated the six possible permutations of the three elements O1, O2, and O3. 

That's what a quotient group is. You get a quotient group whenever there exists a set of functions which is preserved by the action of the group elements...that is, any permutation on the A's, B's and C's returns the same set of functions (the thetas in my example) that you started with. The funny thing is I don't think you'll find it explained this way anywhere. The way it's traditionally done is by defining something called a normal subgroup and its cosets. You investigate the group action on these sets, and you identify the resulting structure as a "quotient group". You can look up quotient groups and normal subgroups on Wikipedia and you'll see that this is how it's done.

Just how different are these two approaches? I think they're very, very different. For one thing, the "normal subgroup" defined in the traditional approach has four elements, while my functional approach operates on three elements. It's different. Different approaches usually give useful insights, but I can't find my method being used anywhere. 

What disinguishes the two approaches is that I treat a group as a set of operations working on a set of target elements, whereas the sophisticated mathematical approach is to ignore the target elements and define everything in terms of the properties of the operations. The formal development of quotient groups uses this lean, efficient methodology. But at what cost?

Sunday, September 16, 2012

What's wrong with Group Theory

I think I figured out what's wrong with group theory. Oh, there's nothing wrong with it; it's all intricately worked out to the ultimate degree. But there's something wrong with the way it's done. Mathematicians are all proud of themselves when they can reduce something to its barest essentials, and that's what they do to group theory.

Group theory has its origins in the theory of permutations. If you have four arbitrary elements, you can re-arrange them in 24 different ways. That's not hard to see. But the interesting thing is that among those re-arrangments, there is a certain structure. For example, the simplest re-arrangment on four elements is perhaps to move each one over one place until you get to the end, and take that one an bring it to the front. If you arranged the elements in a circle, you would just be rotating the whole array one notch. If this is the only operation you permit yourself, then there will be only four possilbe arrangements of your group elements: ABCD, DABC, CDAB, and BCDA.

Another very simple group operation would be to flip the first two elements with each other, leaving the other two where they were. If this is the only operation you allow yourself, then there will be only two possible arrangmenets: ABCD and BACD.

But what if you allow both of those operations? What are the possible outcomes? In particular, using just the flip and the rotation, can you then generate all twenty-four possible permutations of the four letters? That is the kind of question that group theory deals with.

By the way, it's easy to see that the order of operations makes a difference. If you flip first and then rotate, you get DBAC. If you rotate first and then flip, you get ADBC. So the algebra of the group is not, in general, "commutative". Maybe you can see why the Rubik's Cube is subject to analysis by group theory. My own interest is in group theory as applied to algebra of equations: this is the topic of Galois Theory.

Where mathematicians go wrong is they first develop group theory in its most abstract form, developing some very "powerful" theorems along the way. And then they apply those theorems to Galois theory, and prove amazing things about the solvability of equations.

So what's wrong with that? Remember I said they start by developing group theory "in its most abstract form". And one of the manifestations of that abstractness is that they say that the "algebra" of permutation theory doesn't depend on those arbitrary elements which you are shuffling about. (I put "algebra" in quotes to distinguish it from the ordinary meaning of algebra as applied to numbers.) We can develop the whole theory just by looking at the relation between our operations, and ignoring the "things" that they are operating on.

For example, look at our flips and rotations, which we can call F and R. If we flip twice in a row, we get back where we started. So we can say F^2=1:

 Similarly, if we rotate four times, we get back to where we started. So we can say R^4=1:
We need one more fact before we can proceed...something to connect F and R. By fiddling around, we can observe that if we rotate, flip and rotate again, it's the same as flipping, rotating backwards (the same as rotating forward three times) and then flipping. In other words:

From these three relationships we can develop the complete algebra of the group, without any further reference to shuffling about the letters A,B,C,D. For example, using only these relationships we can demonstrate the existence of an element O = RF such that O^3 = 1:

To reduce this equation, we can think pictorially and remember that flipping something twice in a row just brings us back where we started: or we can just use the formal relationship F^2=1 which we took as one of the defining properties of our group algebra. Either way:
So purely by formal manipulations, we have shown that there is a group element which returns our original configuration when applied three times in a row. If we look at our pictorial representation, we can see why. If we take the letters ABCD, rotate and then flip the first two, we get the configuration ADBC: in other words, A stayed where it was, and the letters BCD rotated clockwise. Obviously, three rotations of the letters BCD will return us to ABCD. But somehow, it seems to be more in the spirit of Group Theory, as it is practised within the academic community, to prove this by formal manipulations of symbols.

Am I being quite fair in this? Aren't mathematicians eager to use pictures and graphic constructions whenever they can in order to make their results more understandable? You would normally think so, but I seem to have stumbled across a glaring exception which arises in the critical overlap between Group Theory and Galois Theory. It relates to something called a Quotient Group, which is normally given a very cryptic definition in formal group theory. The irony is that as it applies to Galois Theory, the whole concept of quotient groups makes total sense when it is defined in terms of the letters A,B,C and D which the group operates on: but this intuitive connection is needlessly ignored because the practitioners of Group Theory take enormous pride in their ability to define everything in terms of the F's and R's, so to speak, while ignoring the ABCD's which were the original motivation.

I'll come back to this when I return.

Wednesday, September 12, 2012

The Galois Group of x^5-2

I made a very bad mistake yesterday when I was talking about the fifth roots of two. I said if you moved the first one to the second one, you had to move the second to the third etc. all around the circle in order that the quotients of successive roots remained the fifth root of unity. My mistake comes down to the use of the definitie article: there is no such thing as the fifth root of unity. What would have been true is if I had said that the quotients of successive terms must all equal a fifth root of unity.

Let me explain. For the sake of convenience, I have labelled the fifth roots of two according to their position in the complex plane:
Drawing them out like this makes it appear as though the real root is somehow special. It's not. We can re-shuffle the roots in any number of ways whereby the real root gets mixed in with the complex roots, and there is no algebraic way to tell which is which. It's just convenient for us to have one special configuration to start off with, and this is the obvious choice. We can number the roots one through five, starting with alpha.

I started off yesterday by saying that you couldn't just exchange the first two roots, because the quotient of succesive roots must equal "the" fifth root of unity. Based on this, I said if you move the first root to the second, then the second must go to the third, etc;
You can see that on the left, succesive quotients are all equal the fifth root of unity; and the same holds true on the right hand side.

My mistake was to ignore that successive quotients on the left hand side must all equal the same fifth root of unity as those on the right. But that would be identifying one of those omegas as special...which it isn't. It could be omega squared or omega cubed....any of the fifth roots of unity would do just as well, as long as they are consistent. So moving the first root to the second doesn't restrict our second move at all. We can, for instance move the first root to the second, and the second to the fifth:

The quotient of the first two terms on the left is omega, and on the right it's omega-cubed. But that's long as we make sure all the subsequent quotients on the right are the same:

It turns out wherever you send the first two elements...even if you just swap them with each can always preserve the constancy of the successive quotients by making the correct placements for the final three. Since there are five choices for where the first one goes, and four choices for the second move, there are exactly twenty permissible re-shufflings of the fifth roots of two.

I should point out that I haven't really shown that all these permutations are preserve algebraic consistency. What I've shown is that they don't blatantly violate consistency in any obvious way. It turns out, although I won't show it here, that these permutations are in fact algebraically sound, and the permutiation group as a whole is therefore the Galois Group of the splitting field of x^5-2=0.

Tuesday, September 11, 2012

How can you shuffle the fifth roots of two?

In my last post, I talked about how you can generate new types of numbers by taking square roots (or higher roots) of numbers you already have. You can try and think of more intricate things you might do, like complicated expressions involving sums and quotients, but it turns out that it is sufficient to worry only about expressions where you take roots of things you already have.

On the other hand, the universe of algebraic numbers which actually exist is made up of those numbers which are the solution of algebraic equations. The question becomes: can the members of this set be generated through our constructive mechanism of adjoining roots to fields which we have already created by adjoining roots to smaller fields? We will try to answer this question by looking at the permutation properties of algebraic numbers.

We can show that in the most general case, the roots of a fifth degree equation are indistinguishable from one another: you can swap one for another any which way, and any statement which started out being true will still be true after an arbitrary reshuffling. Of course, we can also find particular fifth-degree equations for which this fails to hold true. But if we are to hope to solve a general fifth-degree equation, then we must expect that we will be able to write an expression which takes on five different values; and that we are furthermore free to shuffle those values about with impunity, always yielding consistently true or false results when resolved to a rational expression.

We remarked last time that for the case of the third degree equation, the three cube roots of two met this requirements. The truth or falsehood of an algebraic expression would not be altered by any arbitrary re-shuffling of the cube roots of two. We have to ask: do we get a similar situation with the fifth roots of two?

We do not. Here is a map showing the location of the fifth roots of two in the complex plane:

You can see that I have taken alpha to be the real fifth root of two, and omega to be a fifth root of unity. You remember when we talked about the cube roots of two, I said we could swap them about any which way and not encounter any contradictions. Well, it doesn't quite work that way with the fifth roots. Suppose we swap, for example, the the two "positive" complex roots, which I'll abbreviate in text as w1 and w2. Now, before swapping them, I could have written:
It's true because the quotient of any two consecutive roots is just omega. But it's no longer true if I swap w1 and w2:

So whatever may be the permutabilities of the fifth roots of two, they are not without some restriction. In fact, from the present example, we can see quite clearly that if we change w1 to w2, the following changes must all follow in due course in order that the quotients maintain their equality:

If you apply this permutation five times in a row, you get back to the original arrangement. By the way, it's important to point out that I haven't shown that this is an allowed permutation of the roots. It's possible that the permutation shown here leads to inconsistent results when you plug the altered roots into some particular equation. All I've shown is that this permutation is not specifically ruled out by the taking of quotients.

Although I'm not going to prove it today, it will turn out that this cycle of permutations is indeed valid for the fifth roots of two. But these are not the only permissible re-shufflings. It goes without saying that complex conjugates can always be swapped with each other. So we can flip the whole circle over the horizontal axis. When we compound this with the rotations, that gives us a total of ten reshufflings. I believe that this is called the "dihedral group on five elements".

Does that complete our description of the Galois group of this particular algebraic field? Not quite. There are other allowed permutations, and they are inherited from the permutation properties of the fifth roots of unity. But we'll take that up another day.