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.

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 OK...as 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 other....you 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.

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.

Shuffling the Roots

I've been writing about the theory of equations and how group theory relates to the question of the solvability of polynomials. I'm trying to do it without falling back on the specialized vocabulary of group theory because I want to see what's actually happening. I took a course many years ago where we proved that you can't solve a fifth degree equation, but I never understood the proof. It was too abstract. I want to understand what is actually happening.

There is one enormously important idea that goes through all of this, and it is what I am going to call the idea of being able to shuffle the roots of an equation. The most obvious example is a quadratic equation with two complex roots. There is no mathematical sense in which we can tell those roots apart. They are different numbers, but anything you say about one is equally true of the other.

What is not so obvious is that the same is true, at least in an algebraic sense, about a quadratic equation with two real (but irrational) roots. Even if one root is positive and one is negative, that doesn't distinguish them algebraically. We can flip them with each other and any true statement algebraic statement about either or both of them (e.g. "they add up to 7") remains true before and after flipping them.

The situation gets even more interesting when we go to the third degree equation. Consider the three cube roots of two. It's not surprising that any true algebraic statement involving the two imaginary roots remains true after swapping them with each other. But surely the real root is special...or so you might think. In fact, it's not so. You can switch the real root with either of the complex roots, and the truth of any algebraic statement about those roots will remain unchanged. You can switch any two of the three, or you can rotate all three in either direction. In other words, you can arbitrarily shuffle the roots. All permutations of the three roots preserve the algebraic properties of the "field", as it is called.

Can we arbitrarily reshuffle the roots of any cubic equation with impunity? There is an obvious counterexample with the three cube roots of eight. One of those roots is just the ordinary number 2, which is clearly distinguishable from the complex roots. But that's obviously related to the fact that you can factor the equation x^3 - 8 = 0. What about irreducible equations?

It turns out you can find irreducible cubics for which not all re-shufflings of the roots are permissible. You get them by building in some structure among the roots. Let's call the roots alpha, beta, and gamma and impose the condition that

If there is to be any kind of symmetry among the three roots, then it's not hard to believe that from the preceding, we must also accept the following cyclic relationships:

It is not too hard to show that from these three equations, we can construct a cubic (actually I think you can make two different cubics) whose roots are alpha, beta, and gamma; and with these three roots, if you substitue alpha for beta in any true equation, you must subsequently, to preserve consistency, subsitute beta for gamma and gamma for alpha. That is, you cannot just swap alpha and beta and leave gamma where it was. You are not permitted to arbitrarily re-shuffle the roots.

In the language of group theory, the permitted re-shufflings of these roots are called the "cyclic group on three elements". In the previous example, where all permutations were allowed, we saw the so-called "symmetric group on three elements". The set of numbers generated by all possible algebraic manipulations of the roots of an equation is called the "splitting field" of that equation, and the group which describes the permissible re-shufflings of those roots is called the "Galois group" of that field. For irreducible cubics, it turns out that these two groups we've described so far (the cyclic and the symmetric) are the only Galois groups that occur "in nature", so to speak.

In algebra, you can define a group of numbers by writing an equation; the numbers which solve that equation are the "roots" of that equation. But such a definition is not especially constructive. It turns out that in a constructive sense, all we can do start with a rational number and take a root (square root, cube root, or whatever)....then consider the field generated by all algebraic combinations of the rationals with that new number (or numbers, in the case of multiple roots). Once we're there, we can do the same thing again....take the root of a number in our new field, and generate a more intricate field by all algebraic combinations of our new and old numbers. And so on.

The big question in algebra is: using this constructive technique, can we generate all the possible numbers which can be defined as the roots of polynomial equations? Of course it will turn out that we can't, and the fifth degree equation will provide our first counter-example. But why can't we? That is what we want to understand.