Why Beauty is Truth Read online

  The normal subgroups of the group of all six permutations of three symbols are the entire group [I, U, V, P, Q, R], the subgroup [I, U, V], whose table we saw just now, and the subgroup with just one permutation, [I]. The other three subgroups, which contain two permutations, are not normal.

  For instance, suppose we want to solve the general quintic. There are five roots, so the permutations involve five symbols. There are precisely 120 such permutations. The coefficients of the equation, being fully symmetric, have a group that contains all 120 of these. This group is the trunk of the tree. Each root, being totally asymmetric, has a group that contains just one permutation—the trivial one. So the tree has 120 leaves. Our aim is to join the trunk to the leaves by branches and twigs whose structure reflects the symmetry properties of the various quantities that arise if we start working out the bits and pieces of a formula for the roots, which we assume are expressed by radicals.

  Suppose for the sake of argument that the first step in the formula is to adjoin a fifth root. Then the group of 120 permutations must split into five pieces, each containing 24 permutations. So the tree develops five branches. Technically, this branching must correspond to a normal subgroup of index 5.

  However, Galois could prove, merely by calculating with permutations, that there does not exist such a normal subgroup.

  Very well, perhaps the solution starts with, say, a seventh root. Then the 120 permutations must split into seven blocks of equal size—but they can’t, because 120 is not divisible by 7. No seventh roots, then. In fact, no prime roots except 2, 3, and 5, because those are the prime factors of 120. And we’ve just ruled out 5.

  A cube root to start with, then? Unfortunately not: the group of 120 permutations has no normal subgroup of index 3.

  All that’s left is a square root. Does the group of 120 permutations have a normal subgroup of index 2? Indeed it does, precisely one. It contains 60 permutations, and is called the alternating group. So by using Galois’s theory of groups, we have established that any formula for solving a general quintic must start with a square root, leading to the alternating group. The first place where the trunk splits leads to just two branches. But there are 120 leaves, so the branches must split again. How do the branches split?

  The prime divisors of 60 are also 2, 3, and 5. So each of our new branches must split into two, three, or five twigs. That is, we must either adjoin another square root, a cube root, or a fifth root. Moreover, this can be done if and only if the alternating group has a normal subgroup of index 2, 3, or 5.

  But does it have such a normal subgroup? That is a question purely about permutations of five symbols. By analyzing such permutations, Galois was able to prove that the alternating group has no normal subgroups at all (except for the whole group and the trivial subgroup [I]). It is a “simple” group—one of the basic components out of which all groups can be constructed.

  Galois’s proof that the quintic is unsolvable.

  There are too few normal subgroups to connect the trunk to the leaves by means of splitting into a prime number of branches at each successive step. So the process of solving the quintic by radicals grinds to an abrupt halt after that first step of adjoining a square root. There is nowhere else to go. No tree can climb from the trunk all the way up to the leaves, and therefore there is no formula for the roots in terms of radicals.

  Using groups to solve the quadratic, cubic, and quartic.

  The same idea works for equations of degree 6, 7, 8, 9—anything larger than 5. This leaves us wondering why the quadratic, cubic, and quartic are solvable. Why are degrees 2, 3, and 4 exceptional? In fact, group theory tells us exactly how to solve the quadratic, cubic, and quartic. I’ll leave out the technicalities, and just show you the trees. They correspond precisely to the classical formulas.

  Now we begin to see the beauty of Galois’s idea. Not only does it prove that the general quintic has no radical solutions, it also explains why the general quadratic, cubic, and quartic do have radical solutions and tells us roughly what they look like. With extra work, it tells us exactly what they look like. Finally, it distinguishes those quintics that can be solved from those that can’t, and tells us how to solve the ones that can.

  The Galois group of an equation tells us everything we could possibly wish to know about its solutions. So why did Poisson, Cauchy, Lacroix, and all the other experts not leap with joy when they saw what Galois had done?

  The Galois group has a terrible secret.

  The secret is this. The easiest way to work out the group of an equation is to use properties of its roots. But of course, the whole point is that we usually do not know what the roots are. Remember, we are trying to solve the equation, that is, to find its roots.

  Suppose someone presents us with a specific quintic, say

  x5 – 6x + 3 = 0

  or

  x5 +15x + 12 = 0

  and asks us to use Galois’s methods to decide whether it can be solved by radicals. It seems a fair question.

  The dreadful truth is that with the methods available to Galois, there is no way to answer it. We can assert that most likely, the associated group contains all 120 permutations—and if it does, then the equation can’t be solved. But we don’t know for sure that all 120 permutations actually occur. Maybe the five roots obey some special constraint. How can we tell?

  Beautiful though it may be, Galois’s theory has severe limitations. It works not with the coefficients but with the roots. In other words, it works with the unknowns, not with the knowns.

  Today, you can go to a suitable mathematical website, input your equation, and it will calculate the Galois group. We now know that the first equation above is not solvable by radicals, but the second one is solvable. My point is not the computer, but that someone has discovered what steps should be taken to solve the problem. The great advance since Galois in this area was working out how to compute the Galois group of any given equation.

  Galois possessed no such technique. It would take another century before routine calculation of the Galois group became feasible. But the absence of this technique let Cauchy and Poisson off the hook. They could complain, with complete justification, that Galois’s ideas did not solve the problem of deciding when a given equation could be solved by radicals.

  What they failed to appreciate was that his method did solve a slightly different problem: to work out which properties of the roots made an equation solvable. That problem had an elegant—and deep—answer. The problem they wanted him to solve . . . well, there is no reason to expect a neat answer. There just isn’t a tidy way to classify the solvable equations in terms of easily computed properties of their coefficients.

  So far, the interpretation of groups as symmetries has been somewhat metaphorical. Now we need to make it more literal, and that step requires a more geometric point of view. Galois’s successors quickly realized that the relation between groups and symmetry is much easier to understand in the context of geometry. In fact, this is how the subject is usually introduced to students.

  To get a feeling for this relationship, we’ll take a quick look at my favorite group: the symmetry group of an equilateral triangle. And we’ll finally address a very basic question: What, exactly, is symmetry?

  Before Galois, all answers to this question were rather vague, handwavy things, with appeals to features like elegance of proportion. This is not a concept you can do sensible mathematics with. After Galois—and after a short period during which the world of mathematics sorted out the general ideas behind his very specific application—there was a simple and unequivocal answer. First, the word “symmetry” has to be reinterpreted as “a symmetry.” Objects do not possess symmetry alone; they often possess many different symmetries.

  What, then, is a symmetry? A symmetry of some mathematical object is a transformation that preserves the object’s structure. I’ll unpack this definition in a moment, but the first point to observe is that a symmetry is a process rather than a thing. Galois�
��s symmetries are permutations (of the roots of an equation), and a permutation is a way to rearrange things. It is not, strictly speaking, the rearrangement itself; it is the rule you apply to get the rearrangement. Not the dish but the recipe.

  This distinction may sound like splitting hairs, but it is fundamental to the whole enterprise.

  There are three key words in the definition of a symmetry: “transformation,” “structure,” and “preserve.” Let me explain them using the example of an equilateral triangle. Such a triangle is defined as having all three sides the same length and all three angles the same size, namely 60°. These features make it difficult to distinguish one side from another; phrases like “the longest side” don’t tell us anything. The angles are also indistinguishable. As we now see, the inability to distinguish one side from another or one angle from another is a consequence of the symmetries of the equilateral triangle. In fact, it is what defines those symmetries.

  Let’s consider those three words in turn.

  Transformation: We are allowed to do things to our triangle. In principle there are lots of things we might do: bend it, turn it through some angle, crumple it up, stretch it like elastic, paint it pink. Our choice here is more limited, however, by the second word.

  Structure: The structure of our triangle consists of the mathematical features that are considered significant. The structure of a triangle includes such things as “it has three sides,” “the sides are straight,” “one side has length 7.32 inches,” “it sits in the plane at this location,” and so on. (In other branches of mathematics, the significant features may be different. In topology, for instance, what matters is that the triangle forms a single closed path, but its three corners and the straightness of its edges are no longer important.)

  Preserve: The structure of the transformed object must match that of the original. The transformed triangle must also have three sides, so crumpling it is ruled out. The sides must remain straight, so bending it is not permitted. One side must still have length 7.32 inches, so stretching the triangle is forbidden. The location must be the same, so sliding it ten feet sideways is disallowed.

  The color is not explicitly mentioned as structure, so painting the triangle pink is irrelevant. It’s not exactly ruled out; it just makes no difference for geometric purposes.

  Turning the triangle through some angle, however, does preserve at least some of the structure. If you make an equilateral triangle out of cardboard, set it on the table, and then rotate it, it still looks like a triangle. It has three sides, they are still straight, their lengths haven’t changed. But the location of the triangle in the plane may still look different, depending on the angle through which you rotate it.

  If I turn the triangle through a right angle, for instance, the result looks different. The sides point in different directions. If you covered your eyes while I turned the triangle, you would know when you opened them again that I had moved it.

  Rotation through a right angle is not a symmetry of the equilateral triangle.

  But if I turned the triangle through 120°, you wouldn’t be able to see any difference between “before” and “after.” To show you what I mean, I will secretly mark the corners with different types of dots, so we can see where it moves to. These dots are for reference only and are not part of the structure that is preserved. If you can’t see the dots, if the triangle is as featureless as any well-behaved Euclidean object, then the turned triangle looks the same as the original.

  Rotation through 120° is a symmetry of the equilateral triangle.

  In other words, “rotate by 120°” is a symmetry of the equilateral triangle. It is a transformation (“rotate”) that preserves the structure (shape and location).

  It turns out that an equilateral triangle has precisely six different symmetries. Another is “rotate by 240°.” Three more are reflections, which turn the triangle over so that one corner remains fixed and the other two exchange positions. What is the sixth symmetry? Do nothing. Leave the triangle alone. This is trivial, but it fits the definition of symmetry. In fact, this transformation fits the definition of symmetry no matter what object we consider or what structure we want preserved. If you do nothing, then nothing changes.

  This trivial symmetry is called the identity. It may seem insignificant, but if we leave it out, the math gets very messy. It’s like trying to do addition without the number zero or multiplication without the number one. If we keep the identity in, everything stays neat and tidy.

  The six symmetries of the equilateral triangle.

  For the equilateral triangle, you can think of the identity as rotation through 0°. On the previous page are the results of applying the six symmetries to our equilateral triangle. They are precisely the six different ways that you can pick up a triangle made of cardboard and lay it down within its original outlines. The dotted lines show where to put the mirror to obtain the required reflection.

  Now I want to convince you that symmetries are a part of algebra. So I will do what any algebraist would do: express everything in terms of symbols. We will name the six symmetries I, U, V, P, Q, R according to the picture above. The identity is I; the other two rotations are U and V; the three reflections are P, Q, and R. These are the same symbols that I used earlier for permutations of the roots of a cubic. There is a reason for this duplication, which will shortly emerge.

  Galois made great play of the “group property” of his permutations. If you perform any two in turn, you get another one. This provides a big hint about what we should do with our six symmetries. We should “multiply” them in pairs and see what happens. Recall the convention: if X and Y are two symmetry transformations, then the product XY is what happens when we first do Y, then X.

  Suppose, for instance, that we want to work out VU. This means that first we apply U to the triangle, then V. Well, U rotates it through 120°, and V then rotates the resulting triangle through 240°. So VU rotates it through 120° + 240° = 360°.

  Oops, we forgot to include that.

  No we didn’t. If you rotate a triangle through 360°, everything ends up exactly where it started. And in group theory it is the end result that matters, not the route taken to get there. In the language of symmetries, two symmetries are considered to be the same if they have the same final effect on the object. Since VU has the same effect as the identity, we conclude that VU = I.

  For a second example, what does UQ do? The transformations go like this:

  How to multiply symmetries.

  We recognize the end result: it is P. So UQ = P.

  With our six symmetries we can form 36 products, and the calculations can be captured in a multiplication table. It is exactly the same table that we obtained for the six permutations of the roots of a cubic.

  This apparent coincidence turns out to be an example of one of the most powerful techniques in the whole of group theory. It originated in the work of the French mathematician Camille Jordan, who arguably turned group theory into a subject in its own right, rather than just a method for analyzing the solution of equations by radicals.

  Around 1870, Jordan drew attention to what is now called “representation theory.” To Galois, groups were composed of permutations—ways to shuffle symbols. Jordan started thinking about ways to shuffle more complicated spaces. Among the most basic spaces in mathematics are multidimensional spaces, and their most important feature is the existence of straight lines. The natural way to transform such spaces is to keep straight lines straight. No bending, no twisting. There are many transformations of this kind—rotations, reflections, changes of scale. They are called “linear” transformations.

  The English lawyer-mathematician Arthur Cayley discovered that any linear transformation can be associated with a matrix—a square table of numbers. Any linear transformation of three-dimensional space, for example, can be specified by writing down a 3-by-3 table of real numbers. So transformations can be reduced to algebraic computations.

  Representation theory le
ts you start with a group that does not consist of linear transformations and replace it with one that does. The advantage of converting the group to a group of matrices is that matrix algebra is very deep and powerful, and Jordan was the first to see this.

  Let’s look at the symmetries of the triangle from Jordan’s point of view. Instead of placing shaded dots in the corners of the triangle, I will place the symbols a, b, c, corresponding to the roots of the general cubic. It then becomes obvious that each symmetry of the triangle also permutes these symbols. For example, the rotation U sends abc to cab.

  The six symmetries of the triangle correspond naturally to the six permutations of the roots a, b, c. Moreover, the product of two symmetries corresponds to the product of the corresponding permutations. But rotations and reflections in the plane are linear transformations—they preserve straight lines. So we have reinterpreted the permutation group—represented it—as a group of linear transformations, or equivalently as a group of matrices. This idea was to have profound consequences for both mathematics and physics.

  How symmetries of the equilateral triangle correspond to permutations.

  8

  THE MEDIOCRE ENGINEER AND THE TRANSCENDENT PROFESSOR

  No longer was symmetry some vague impression of regularity or an artistic feeling of elegance and beauty. It was a clear mathematical concept with a rigorous logical definition. You could calculate with symmetries and prove theorems about them. A new subject was born: group theory. Humanity’s quest for symmetry had reached a turning point. The admission fee for this advance was a willingness to think more conceptually. The concept of a group was an abstract one, several stages removed from the traditional raw materials of numbers and geometrical shapes.