Group Theory and Various Important Sets

Some time ago I became interested in group theory for two different reasons: first, because of the role it plays in defining the unique properties of some common sets of numbers in mathematics, and second, because of the role it plays in physics. The following is an attempt to describe the train of thought involved in the first topic (the unique properties of some important number sets), but I end up with some comments on the second as well. My presentation here is oversimplified and non-rigorous; any readers who already know about this stuff will probably think I'm stating the obvious and doing it pretty clumsily. If you're interested in a more in-depth discussion of what I'm only scratching the surface of here, try this site; the "fundamentals" section talks about the main sets I discuss in this article but with more mathematical background and rigor.

The concept of a "group" will seem pretty abstract when we look at its precise mathematical definition below, but the idea behind it is actually quite simple. Mathematics has to deal with a wide variety of objects and sets of objects, and it's useful to make this as easy as possible by coming up with ways in which different objects and sets of objects are similar, and then abstracting out the similarity and making it into some type of logical system. For example, the set of natural numbers, N, or {0, 1, 2, 3, ...} is an abstract structure that was invented to capture the similarities between all the different sets of things that can be counted. Once we have this set, and understand the logic underlying its abstract structure, we can deduce propositions that will apply to anything that can be described by that abstract structure; we don't have to know what kind of thing it is, or even whether it actually exists. The same goes for all of the abstract structures studied in mathematics. (This is why Bertrand Russell once described mathematics as "the subject in which we do not know what we are talking about, nor whether what we are saying is true.")

Once you've been doing mathematics for a while, you find that there are lots of these different abstract sets lying around, and sooner or later you realize that it's time to take this process of abstraction to the next level. That is, you start to realize that there are abstract similarities between different abstract structures, and you start studying the logic underlying the abstract structure that is common to all these different abstract structures. (Whew!) Group theory is an example of this "second-order" study of abstract structures; it's a way of axiomatizing properties that many important abstract structures (such as, for example, the set of integers Z, the set of rational numbers Q, the set of real numbers R, and the set of complex numbers C) have in common, so that they can be studied independently of the specific sets themselves.

Historically, group theory arose in the 19th century in the course of studying the roots of polynomial equations of various degrees. The particular question at the time was whether there could be a general procedure for finding roots of polynomials of the fifth degree. You probably remember from algebra class the general procedure for finding the roots of quadratic (second degree) polynomials; it is the quadratic formula:

x = ( -b +/- √(b2 - 4ac) ) / 2a

where the coefficients come from the general quadratic polynomial, ax2 + bx + c = 0.

It turns out that this kind of thing also works with cubic and quartic (third and fourth degree) polynomials, but it doesn't work for polynomials of 5th degree and higher. The reason turns out to be connected to the symmetry properties that the set of solutions has to have: in other words, the solutions form a group, and it can be shown that it's a group that can't possibly be the same as the group of symmetries of quantities that can be constructed using the operations that are used to construct the formulas for solving equations of degrees lower than the fifth. (If you're interested, there's a good discussion that gives more detail--and by someone who's an expert in the field, unlike me--on John Baez's web site.)

For a century or so after it was invented, group theory seemed to be one of those pieces of higher mathematics that is beautiful, elegant, and totally unrelated to the real world. (How often does the average engineer need to find solutions to fifth degree or higher polynomials?) However, in the twentieth century, with the development of relativity and quantum mechanics, it turned out that there were various symmetry groups that could be found in nature itself, and understanding them was of crucial importance for the advance of physics. This is often cited as an example of what the physicist Eugene Wigner called "the unreasonable effectiveness of mathematics" in science--some abstract structure that mathematicians discover in the course of investigating some esoteric mathematical problem turns out to be intimately involved in physics.

For now, we are going to stick to the basics, and look at the fundamental axioms and properties of groups. Then we'll take a look at the sets I mentioned above, and see how their properties exemplify various more specific types of groups. (At the end, for "extra credit", I'm going to return briefly to the topic of symmetry groups that are observed in nature.)

Definition. A group is a set G and a binary operation *: G x G acting on members of G (usually called elements of the group), which together satisfy the following (these are often called the group axioms):

(1) Closure. G is closed under *, that is: if a * b = c and a, b are elements of G (we shall henceforth write "in G" for this), then c is in G.

(2) Associativity. The * operation is associative over G, that is: if a, b, and c are in G, (a * b) * c = a * (b * c).

(3) Identity. There is an element e of G, called the identity, such that, for all a in G, a * e = a.

(4) Inverse. Every element a of G has an inverse, denoted a-1, which is also in G and satisfies a * a-1 = e.

Notation. While the operation * may be anything which applies to the elements of the set G, the two most common operations are addition (customarily written a + b for a, b in G) and multiplication (customarily written ab). Note that the group operation, whether it is addition, multiplication, or something else, is not necessarily commutative, so the order in which a, b are written matters. Note also in this connection that I have stated the axioms above with only one ordering. This really only makes a difference with the last two; technically, the axioms only specify that e is a "right identity" (because it's on the right in the axiom) and that a-1 is a "right inverse" (for the same reason). It can be shown from the axioms as I have given them that e is also a left identity and that a-1 is also a left inverse, even in a group where the group operation is not always commutative, but I won't do it here. Groups whose operation is always commutative (i.e., a * b = b * a for all a, b in G) are called abelian groups.

Finite Groups. A group is finite if its underlying set G has a finite number of elements. The number of elements is called the order of G, denoted [G]. We won't be looking much at finite groups in this article since the sets we'll be focusing on (the integers, the reals, etc.) are all infinite sets, but I wanted to mention them because I'll give a couple of examples of them below (just to show that the ideas of group theory don't just apply to infinite sets).

Scalar Multiplication. For any additive abelian group G (i.e., the group operation is addition, and it is commutative, a + b = b + a for all a, b in G), another operation can be constructed as follows: a + a + ... (n times, where n is an integer) is called scalar multiplication of a by n, and is written na. Note that this is not the same as "group multiplication", the multiplication of one element of a group by another, which we will discuss next.

Rings. A ring is an additive abelian group which also has a multiplication operation defined that satisfies (1), (2), and (3) above, but not necessarily (4), and which is distributive over addition. That is, the following hold for a ring R:

(1) R is closed under addition (if a + b = c and a, b are in R, c is in R);

(2) Addition is associative over R;

(3) R has an additive identity, 0, such that a + 0 = a for all a in R;

(4) Every a in R has an additive inverse in R, -a, such that a + -a = 0;

(5) Commutativity of Addition. For all a, b in R, a + b = b + a;

(6) R is closed under multiplication (if ab = c and a, b are in R, c is in R);

(7) Multiplication is associative over R;

(8) R has a multiplicative identity, 1, such that a1 = a for all a in R;

(9) Distributivity of Multiplication over Addition. For all a, b, c in R, (a + b)c = ac + bc, and also a(b + c) = ab + ac.

(Food for thought: if every element a has to have an additive inverse -a, what is the additive inverse of 0?)

Technically, the above axioms say that R is an abelian group with respect to addition, but it is only a monoid with respect to multiplication. (Note: Some definitions I've seen only have a ring obeying the first two group axioms for multiplication, not the first three--under this definition rings have no multiplicative identity. A ring that also has a multiplicative identity is then called a "unit ring". This would make a non-unit ring only a "semi-group" with respect to multiplication; lack of an identity element is what differentiates a semi-group from a monoid. Since all of the rings we'll be looking at here do have a multiplicative identity, I'm going to ignore this technicality for the rest of this article.)

The best-known example of a ring is, of course, the set of integers, usually denoted by Z. In fact, since multiplication is also commutative over Z, Z is an abelian or commutative ring. There are also, of course, non-commutative rings; for example, the set of all n x n matrices with integer entries for a given n, with the normal matrix addition and multiplication operations.

(Food for thought: what about the set N of natural numbers? Is it a ring? Is it a group?)

There are also many finite subsets of the integers which are rings: in fact, the set of integers modulo n, where n is any integer greater than 1, form a ring. (The technical name for this type of set is Zn.) For example, consider Z3 = {0, 1, 2}; it's closed under addition and multiplication (because 1 + 2 = 0, since 3 is 0 modulo 3, and 2 + 2 is 1--multiplication should be obvious because 2 x 2 is the same as 2 + 2), and you can see that all the other ring axioms are also satisfied. (We'll look at Z3 again in a moment because in fact it also satisfies a couple of other important axioms, which we'll see next.)

Division Rings. A special type of ring is the division ring, which is a ring that also has a multiplicative inverse. That is, a division ring R, in addition to satisfying (1) through (9) above, also satisfies the following:

(10) Every a in R\0 (the set of all elements of R except 0) has a multiplicative inverse in R\0, a-1, such that aa-1 = 1.

(Food for thought: what is the multiplicative inverse of 1? Should be easy if you answered the analogous question for addition above.)

Note that the integers Z are not a division ring (why not?). To get a division ring, we must extend the underlying set so that each element has a multiplicative inverse satisfying (10). Equivalently, we must form the closure of Z under the operation of division, the inverse of multiplication. (The integers Z themselves are the closure of the set N of natural numbers under subtraction, the inverse of addition.) The result when we so extend Z is the set Q of rational numbers, which are all numbers that can be expressed as a ratio of integers, p / q. We note that multiplication is commutative over Q, so that it is actually a more specific type of division ring, called a field.

Fields. A field is a division ring for which multiplication is commutative; that is, in addition to (1) through (10), a field F satisfies the following:

(11) Commutativity of Multiplication. For all a, b in F, ab = ba.

Several well-known sets in mathematics are fields: the set Q of rational numbers, mentioned already, the set R of real numbers, and the set C of complex numbers. All of these fields are unique in the sense that any field which has the particular set of properties that makes them unique must be isomorphic to them. (I won't define the term isomorphic explicitly here, but intuitively, two sets are isomorphic if there is a complete one-to-one correspondence between them that "preserves structure"--leaving that phrase undefined for now, but that gives you an idea of what's going on.) Most of the rest of this article will be devoted to examining the special properties of each of these fields that make them unique.

All of these sets are infinite, but just as with groups and rings above, there are also finite fields. In fact, the set Z3 that we looked at above is an example: it should be obvious that multiplication is commutative, and if you think about it a little, you will see that each nonzero element also has a multiplicative inverse (which is the key property that makes it a division ring instead of just an ordinary ring). In fact, any set Zn where n is a prime number is a field. (What about Zn when n isn't prime? These aren't fields because some elements--in fact, any which are factors of n--will not have multiplicative inverses, because they are "zero divisors": they give zero when multiplied by some member of the set, since n is zero modulo n. You should be able to convince yourself easily that a zero divisor can't have a multiplicative inverse--though note that the converse is not true: there can be rings which have no zero divisors but which also do not have multiplicative inverses for all elements. A commutative ring with no zero divisors is called an integral domain, because the integers Z are the best-known example of one--and they also illustrate that there can be integral domains which are not division rings, as I stated just now.)

Ordering. A partial ordering on a set S is a relation \ satisfying the following properties:

(i) Reflexivity: a \ a for all a in S;

(ii) Antisymmetry: If a \ b and b \ a, then a = b for a, b in S;

(iii) Transitivity: If a \ b and b \ c, then a \ c for a, b, c in S.

Note that this set of properties is very similar to the three properties satisfied by an equivalence relation; the only difference is property (ii), which for equivalence relations becomes the property of symmetry (if a = b, then b = a). The antisymmetry property is what makes the relation an ordering relation rather than an equivalence relation.

Note also that there is no guarantee that for every pair of elements a and b of S, at least one of a \ b or b \ a is true. There may be some pairs of elements for which neither is satisfied (such pairs of elements are called incomparable). This is why the relation \ is called a partial ordering. If it is true that every pair of elements can be ordered, then the relation is called a total ordering of S--this is sometimes expressed as a fourth property which is satisfied by totally ordered sets (but not partially ordered sets):

(iv) Trichotomy: For all a, b in S, at least one of a \ b or b \ a is true.

This fourth property is called "trichotomy" (even though as I've stated it here it involves only two "options") because it is possible that both a \ b and b \ a are true (which is then the third "option"), in which case by antisymmetry a = b. Sets which have a total ordering are called (reasonably enough) totally ordered sets. (The term "ordered sets" is often used to mean totally ordered sets, but it is sometimes used to include partially ordered sets as well.)

The field Q of rational numbers is a totally ordered set; in fact, there are two basic orderings that can be defined, less than or equal to (ab) and greater than or equal to (ab). (Note that a < b and a > b, less than and greater than, are strictly speaking not total orderings, since neither holds if a = b. We will ignore this technicality in what follows, and write b < a and b > a freely, not worrying about possible cases where b = a.) However, ordering alone is not enough to uniquely define Q, since the field R of real numbers is also ordered. So what other properties might serve to distinguish Q?

Density. An ordered set S is dense if, given elements a and b such that ab, there exists some element c which lies between a and b. That is, there exists an element c such that:

(i) ca and cb;

(ii) If a < b, then a < c and c < b;

(iii) If a > b, then a > c and c > b.

It is easy to see that Q is dense: for any elements a and b such that a < b, we simply take c = (a + b) / 2. However, this trick will also work for R, so again the density property is not enough to uniquely define Q. (We note, however, that the rationals Q are also dense in the reals R, and so are the irrationals I: that is, between any two elements of Q there is an element of R which is not in Q--and hence is in I--and between any two elements of I there is an element of Q. Q and I are thus disjoint, relatively dense sets which together make up R.)

We won't discuss the density property further in this article, but it is worth an aside to note that the density property, applied to the reals R, is part of Cantor's definition of the continuum, which was a watershed in the history of mathematics. For a couple of centuries mathematicians had been using the calculus, which had been invented by Newton and Leibniz, without having really rigorous foundations for some of its basic concepts; Cantor's definition of continuity provided the key to constructing such a rigorous foundation. (It also provided, for the first time, a way of avoiding the paradoxical conclusions about motion which were enunciated by Zeno in the fifth century B.C., but that's a topic for the philosophy section.)

Well-Ordering. A set S is well-ordered if there is an ordering on S such that every non-empty open subset of S has a smallest element, which is an element a such that there is no element b for which b < a.

The set Q of rational numbers is well-ordered. We know this because Q is countable, i.e., it is capable of being placed into a one-to-one correspondence with the set N of natural numbers. All countable sets are well-ordered; the well-ordering is given by the one-to-one correspondence with N, and the smallest element of any open subset is simply the element corresponding to the smallest natural number. A well-ordering of the entire set Q will have as its smallest element the number put into correspondence with the number 0. However, this example also shows that a well-ordering of a set cannot always use the "standard" ordering of the set, since any method of putting Q into a one-to-one correspondence with N will have to use an ordering that isn't the same as the "standard" ordering on Q, the one given by ≤ and ≥ that we saw above. (Food for thought: why is this true? Answering this will show why we can't construct a well-ordering of Q using the "standard" ordering. Think about the density property above. This also means that a well-ordering of a set may not be compatible with the operations that make it a field, a ring, or even a group.)

The question of whether R is well-ordered is more complicated. It turns out that this question depends on the question of whether the Axiom of Choice (AC) holds. The Axiom of Choice says that, given any collection of non-empty sets, there is a set which contains exactly one element from each member of the collection. AC seems obvious and innocuous enough when stated, but it turns out that, when combined with the standard Zermelo-Fraenkel axioms of set theory (ZF--the combined system is usually referred to as ZFC), it implies a number of counterintuitive results, one of which is that all sets, including R, must be well-ordered. However, AC gives us no idea how to well-order a given set, such as R, and it is very difficult to see how one could possibly well-order a set like R which is uncountable (see below). This is one of the objections to accepting AC without qualifications, at least in its usual form. (Mathematicians can, and do, still investigate all the implications of the combined system ZFC, but because of the issues that arise with AC, they also look at alternative axioms that yield different sets of theorems, some of them contradictory to theorems of ZFC. Since, as we've seen, some of the theorems of ZFC are highly counterintuitive, the alternative systems can look attractive for some types of applications. What all this boils down to is that, contrary to what most people initially expect, there is no one unique axiom system that adequately captures all of our intuitions about sets--or indeed about any mathematical objects. Many people don't like this, and when it was first discovered in the twentieth century some mathematicians darkly predicted the doom of the entire field, but one learns to cope with these things.)

So we can't use well-ordering either as a property to distinguish Q from R; but in our discussion we've identified another property which we can use to do so: countability. Cantor first showed in the late 1800's that R is not countable, using a very slick argument called the diagonal argument, which has turned out to be a very useful general trick for getting important results in mathematics. I won't describe the argument here, but Googling on "cantor diagonal argument" will turn up a number of good discussions of it. The upshot of the argument is that no matter how you try to construct a one-to-one correspondence between N and R, you can't do it--you will always end up leaving out an infinite number of reals, and therefore the reals are not countable. So countability is the crucial feature that distinguishes Q from R (and all other ordered fields), and hence enables us, along with the other properties we have listed, to uniquely define it:

There is a unique ordered field which is countable, and it is the set Q of rational numbers.

Of course this immediately raises the question: if Q is the unique ordered field which is countable, is R just the corresponding unique ordered field which is not countable? Unfortunately, no; at the very least, there are other uncountable proper subsets of R which are also ordered fields, so we have to look further to find a property that uniquely picks out R itself.

Sequences and Bounds. A sequence is a countable subset of an ordered set S with no two elements equal. A bounded sequence B is a sequence which satisfies at least one of the following two conditions:

(i) If there is an element u of S such that, for every a in B, au, then B is said to be bounded above, and u is an upper bound for B;

(ii) If there is an element l of S such that, for every a in B, al, then B is said to be bounded below, and l is a lower bound for B.

A given sequence may have many upper or lower bounds. For example, consider the sequence in Q defined by {a(n): an = 1/n, n in N, n > 0}. This sequence has 0 as a lower bound and 1 as an upper bound; but it also will have any number less than 0 as a lower bound and any number greater than 1 as an upper bound (since all such numbers will satisfy the appropriate condition above). Note also that a bound may be a member of the sequence, but does not have to be; for the above sequence, 1, an upper bound, is the member a1, but 0, a lower bound, is not a member.

There is, of course, something special about 0 and 1 for the above sequence, and it is captured by the following pair of definitions:

(i) An element u of S is said to be a least upper bound for a sequence B in S if there is no upper bound v for B for which v < u;

(ii) An element l of S is said to be a greatest lower bound for a sequence B in S if there is no lower bound m for B for which m > l.

Obviously, for the sequence given above, 1 is the least upper bound, and it only takes a little more thought to see that 0 is the greatest lower bound. Finding least and greatest bounds tells us that we have "pinned down" a sequence to the narrowest possible range in its parent set S. (A bit of terminology: a least upper bound is often called a supremum--abbreviated "sup"--and a greatest lower bound is often called an infimum--abbreviated "inf".)

A question arises, however: is it always the case that any sequence in a set S which has an upper bound in S will have a least upper bound in S? The answer, perhaps surprisingly, turns out to be no. In fact, the set Q of rational numbers provides a counterexample: there are sequences in Q which have upper bounds in Q, but which do not have a least upper bound in Q! The shorthand way of saying this is that Q does not have the least upper bound property. (We could also ask the converse question about lower bounds and greatest lower bounds, of course; however, it turns out that one property implies the other--any ordered set with the least upper bound property will also have the converse greatest lower bound property. So we only need to talk about one of the two in our discussion here.)

Let us look at an example. Consider the following sequence in Q: {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}. You will probably recognize this sequence as successive truncations of the decimal expansion of √2. The squares of these numbers will get closer and closer to 2 from below, without ever quite reaching it. Now let's pick out some upper bounds for the sequence. We know that 1.5 is an upper bound, because the square of 1.5 is 2.25, which is greater than 2. But we can find a smaller upper bound: 1.42. Its square is 2.0164, still greater than 2. We can find a still smaller upper bound: 1.415. Its square is 2.002225. We can find yet a smaller...but you get the idea. No matter what upper bound we choose in Q, we will always be able to find a smaller number in Q which is also an upper bound; and this is because the least upper bound for the sequence, which is simply √2 itself, is not in Q. The ancient Greeks proved this as a theorem, and it threw them into consternation for a long time, until a few mathematicians (notably Eudoxos) came along and produced an axiomatic system for dealing with what we now know as real numbers.

The least upper bound of the sequence we just examined, √2, is of course in R, so if we view it as a sequence over R instead of Q, then R has the least upper bound property with respect to that sequence. It takes quite a bit more work to prove that R has this property with respect to every sequence over itself. It turns out that this is the distinguishing feature of R: that it has the least upper bound property. That is,

There is a unique ordered field with the least upper bound property, and it is the set R of real numbers.

You might wonder why the least upper bound property is so important. There are a number of reasons, but the primary one that motivated mathematicians to precisely define the reals was the desire to put the theory of limits, and thereby calculus, on a sound axiomatic footing. Without the least upper bound property, it would not be possible to ensure that all sequences have definable limits, which means that it wouldn't be possible to ensure that derivatives and integrals, the fundamental tools of calculus, were always well-defined.

We now have a unique combination of properties that picks out the reals. But is this the end of the line? That is, are the reals the "ultimate" mathematical set, which has all of the properties we might want? The answer, perhaps surprisingly, turns out to be "no"--there are properties that are "nice" for a set to have that the set of reals doesn't have. Let's look at the main one.

Polynomials and Roots. A polynomial over a set S is an expression of the form P(x) = Σ (i = 0 to n) aixi, ai in S for all ai, x in S (but x is treated as a variable, whereas all of the ai are treated as constants), and i, n in N. The highest power n of x that appears in the polynomial is called the degree of the polynomial. (Note that we are assuming that an, the coefficient of xn, is nonzero; otherwise we would simply reduce n, the upper limit of the sum. Other coefficients of lesser powers of x may be zero.) A root of a polynomial is a value of x for which P(x) = 0.

The basic question in the study of polynomials is whether a polynomial over a set S must have at least one root in S. A well-known counterexample shows that this is not the case for Q or R: the polynomial x2 + 1 has all its coefficients in Q (and hence R), but has no roots in R (and hence none in Q); that is, the equation x2 + 1 = 0 has no real solutions. It was equations like this one which helped to spur the extension of the number system to include imaginary numbers. You are probably familiar with imaginaries, but let us take it slowly for a bit and introduce them axiomatically.

Let i be a number such that i2 = -1. Obviously i cannot be a real number; it must be some new type of number which is not included in the set of reals. However, we can as a working hypothesis assume that i can be combined with real numbers by the usual arithmetic operations, and see where that leads us. Thus, we obtain the following:

(i) Any number of the form bi, with b in R, is called a pure imaginary number. The squares of pure imaginary numbers are negative real numbers: (bi)2 = -b2. The pure imaginaries have a number of other special properties with certain functions, in particular the exponential ex, which we will explore below. Note that if b = -1, bi = -i, and the square of this number is also -1 (thus -1 has two square roots).

(ii) Numbers of the form z = a + bi, with a, b in R, are called complex numbers. Complex numbers obey the same rules of algebra as real numbers, with the added proviso that i2 = -1. It is easy to show that the set C of complex numbers, so defined, obeys all of the axioms (1) through (11) above that characterize a field. The real number a is called the real part of z, and the real number b is called the imaginary part of z.

(iii) The number z* = a - bi is called the complex conjugate of z.

(iv) The modulus |z| of a complex number z is the positive real number r given by r2 = zz* = a2 + b2, where a and b are the real and imaginary parts of z.

(v) Any complex number z can be written as the product of its modulus r and the exponential eiq, where the real number q is called the argument. If r = 1 we have the subset of C called the unit circle; this terminology arises because the complex numbers can be plotted in a plane, with the x-axis being the reals and the y-axis being the pure imaginaries; the real and imaginary parts of a complex number z are then its x- and y-coordinates. The number q is then the angle (in radians) between a line segment drawn from the origin to the point designating z, and the positive real axis (x-axis), measured counterclockwise; the modulus r is simply the length of this line segment. If r = 1 then the line segment is 1 unit long and the point z lies on a circle of unit radius around the origin.

(vi) If z = reiq, then the complex conjugate z* = re-iq; the point z* lies at the same radius as z from the origin, but with the angle taken in the opposite direction (clockwise).

(vii) Raising z to a power a is now simple: obviously za = raeiaq. Note that, if a is real, this form makes it obvious that if z is in C, za is also in C. It should only take a little more thought to see that even if a is a complex number, za will still be a complex number. This shows that C is closed under the operations of raising to powers and extracting roots (since extracting the ath root is equivalent to raising to the power 1/a), while R and Q were not.

Fundamental Theorem of Algebra. The above propositions are enough to show that the set C is "big enough" to allow us to do what we want with polynomials: we can find a root in C for any polynomial with coefficients in C. This is called the fundamental theorem of algebra, and is one of the most important results in higher mathematics. The proof is rather involved, but the crucial step depends on property (vii) above, that C is closed under the operations of raising to powers and extracting roots.

This property of closure under the process of taking roots of polynomials is one of the unique features which distinguishes the complex numbers. However, it turns out that there are other fields which are also "algebraically closed" (that's the technical name for the property we've just discussed). The key additional distinction here comes from the fact that all those other algebraically closed fields are generated from the rational numbers, Q; there are more than one, in fact an infinite number, because it turns out that there are an infinite number of ways of constructing algebraic closures of Q. The complex numbers, however, are the algebraic closure of the reals, R, and it turns out that there is only one unique way of doing that. So we could say that C is the unique field which is the algebraic closure of the reals, but in the spirit of the above definitions, we'd like to express this uniqueness in a way that doesn't bring in any of our other sets. We can do this by observing that the algebraic closure of any countable field must also be countable--which means that all of those other fields which are algebraic closures of Q are countable. C, by contrast, is obviously not countable (since it has R as a subset), and so we can express C's uniqueness better using its uncountability as the key property. That is,

There is a unique field which is both uncountable and algebraically closed, and it is the set C of complex numbers.

Note that C is also distinguished from R by the fact that C, unlike R, is not ordered: it is only partially ordered. One can impose a variety of partial orderings on C: the most common one uses the modulus (defined earlier) to generate the ordering. The partial ordering induced by the modulus is simply the following: z < w iff |z| < |w|. However, since there are infinitely many complex numbers with a given modulus, this can only be a partial ordering (complex numbers with the same modulus are incomparable under this partial ordering). (Food for thought: why could I not use ≤ instead of < just now? The answer is easy if you think about why numbers with the same modulus are incomparable.) The fact that C is partially ordered, by the way, also explains why all those fields which are algebraic closures of Q aren't counterexamples to the unique definition of Q (as the only countable ordered field) that we gave above: all of those algebraic closures of Q are (countable) subsets of C but not of R (because they have to include imaginary elements in order to provide negative rational numbers with roots), so they aren't ordered the way Q is.

Quaternions and Octonions. I can't resist a digression here to talk about the two "hypercomplex" sets, quaternions and octonions, which share with the complex numbers the property that all polynomials over the set have roots in the set. (This does not contradict the uniqueness of C as defined above, because neither the quaternions nor the octonions are a field--the quaternions are non-commutative with respect to multiplication, so they are only a division ring, and the octonions are non-commutative and also non-associative, so they are not even, technically speaking, a group--they are a "division algebra", which is a set that has a multiplication operation with an inverse, but not all division algebras are groups because they're not all associative. It's harder, for me at least, to intuitively grasp how a set can be non-associative under multiplication than to grasp how it can be non-commutative, but it turns out that the non-associativity of the octonions has some interesting consequences relative to other groups that are important in physics. But I'm digressing from the digression.)

The quaternions were first discovered by the Irish mathematician William Rowan Hamilton in the nineteenth century, in the course of other investigations. As with group theory in general, nobody had the slightest idea that quaternions would ever have any use in the real world, but in the 1920's when quantum mechanics was being developed, Pauli (who was unaware of Hamilton's work) basically re-invented the quaternions as what are now called the "Pauli spin matrices" because they come into play when constructing wave functions for the "spin" degrees of freedom of elementary particles. (The matrices that Pauli used are actually the "Hermitian counterparts" of the quaternions that Hamilton used, but I won't digress from the digression again to explain what that means, except to say that they have fundamentally the same structure, they're just written in a somewhat different way because of the particular requirements of quantum mechanics.)

The idea behind the quaternions is simple to state, though it sounds wacky: instead of there just being one "imaginary unit" i, as in the complex numbers, suppose that there are three, which we will label i, j, and k. When we say that all three of these things are "imaginary units", we mean that all three of them square to -1. Now of course it was tough enough to accept that -1 had one "square root", but how can it possibly have three? It looks as though all of them are "just the same, only different". Well, however counterintuitive it sounds, it turns out to be possible, in the sense that a fully self-consistent division algebra can be set up using the four numbers, 1 and the three imaginary units, as "basis vectors". (It would be another digression from the digression to explain that term, since I haven't talked at all about vector spaces here--another article, maybe?--but for our purposes now suffice it to say that 1 and i are basis vectors of the complex numbers C in the same sense.) All we need is to define the "multiplication table" for the three imaginary units, and we can then add and multiply any two quaternions, and prove that those two operations satisfy all the axioms of a division algebra. (Food for thought: why do we not need an "addition table" for the imaginary units? Answering this will give you a better idea of what "basis vectors" means.)

Defining the multiplication table is easy: we simply say that, by definition, ij = k, and then generate the rest of the table by permuting this equality, with the sign of the right-hand side remaining positive for even permutations but becoming negative for odd permutations (so, for example, jk = i, but ji = -k). Since the division algebra set up in this way turns out to be non-commutative (food for thought: can you see that this is "obvious" from the multiplication table I just gave?), it is indeed a different group from the "ordinary" complex numbers, and qualifies as a distinct mathematical structure in its own right.

Once you've bought into this, then defining the octonions is easy: instead of three imaginary units, the octonions have seven (usually denoted e1 through e7), which with the identity 1 makes a total of eight basis vectors for this new mathematical space. (It should now be clear where the names "quaternion" and "octonion" come from--the prefixes just refer to the number of basis vectors in the space.) Since the octonions, defined in this way, are not only non-commutative but non-associative, they again are clearly a distinct mathematical space in their own right.

You may ask, how long can this go on? Can we have a set of "sextadecions" with sixteen basis vectors (the identity 1 plus fifteen--count them--imaginary units)? And so on, ad infinitum? Or, to take another tack, why must the number of basis vectors be a power of two? Why do we not have such sets with three or five or some other number of basis vectors? (Food for thought: we can, if we like, pick out a division algebra sharing a number of properties with these sets--though not all of them or even some of the most important ones--that has only one basis vector. What is it?) Can't we construct similar mathematical spaces with any number of basis vectors we like?

The answer (just to keep everyone guessing) turns out to be no. Of course we can construct general vector spaces with any number of "dimensions" we like, but (like the "one-dimensional" vector space R) they won't have a number of important properties (such as having roots for all polynomials). It turns out (though I'm certainly not going to try to explain why here, since I don't entirely understand it myself) that there are only three possible "hypercomplex division algebras" that do have all of those important properties, and they are the three sets we have just discussed: C, H (the quaternions), and O (the octonions). And again, this abstract mathematical fact turns out to have effects in physics, in particular with the number of possible different "theories of everything" that there are having certain structures, and how many dimensions these theories can have. But I'm getting out of breath, so let's end this article with something that's (comparatively) simple.

The Unit Circle. We've finished the discussion of basic group theory and the properties of the important sets Q, R, and C; but for "extra credit" let's examine the subset of C with r = 1; that is, the set U, the "unit circle", defined as {z: z = eiq, q in R}. It's not hard to establish that U is itself a group under multiplication--that is, it obeys the group axioms (1) through (4) above, with multiplication as the group operation. This group has a special name in group theory: it is called the unitary group of dimension 1, or U(1).

The group U(1) is an example of a Lie group, which is a special type of group that turns out to be very useful in physics. This is because all of the symmetry groups we find in nature are Lie groups. A "symmetry group" is a group whose elements are symmetry transformations--that is, transformations that leave some property of the transformed objects invariant. Symmetry groups in nature are groups of transformations that can be applied to the various quantities that appear in the laws of physics, but which leave the laws of physics themselves unchanged. Now in the laws of physics there are also what are called "conservation laws", which are laws that say that a particular physical quantity (like momentum, energy, etc.) is unchanged in various physical processes. It turns out that each group of symmetry transformations "corresponds" to a particular conservation law (I won't go into the details of why this is true, since it's not necessary for the present discussion, but it's a fascinating topic).

For example, consider a very "easy" group of symmetry transformations, called "translations"--or more precisely, "spatial translations". If you think of a set of spatial Cartesian coordinates (mutually perpendicular axes), then spatial translations simply move the origin of the set of coordinates, without changing the orientation of the axes. It turns out that all of the laws of physics are left unchanged by this transformation of coordinates, and this fact turns out to correspond to the law of conservation of momentum--more precisely, the fact that the laws of physics are invariant (unchanged) under spatial translations explains why there is a law of conservation of momentum. Similarly, the fact that the laws of physics are invariant under "time translations" (changing the "zero point" of time, like resetting your watch) corresponds to the law of conservation of energy, and the fact that they are invariant under spatial rotations corresponds to the law of conservation of angular momentum.

So how does the group U(1) fit into this picture? What symmetry transformations are elements of U(1)? It certainly doesn't look like they are anything simple like time or space translations, or rotations. And they're not. It turns out that the classical laws of electromagnetism, Maxwell's Equations, exhibit a symmetry called (for historical reasons) a "gauge symmetry". The name really doesn't tell you much about how this symmetry works, but the idea is that you can take the fields that appear in Maxwell's Equations and multiply them by elements of U(1) (i.e., complex numbers of the form eiq), and the equations themselves are left unchanged by this transformation--in other words, the laws of electromagnetism are invariant under these "gauge transformations" of the fields. Now this fact also corresponds to a conservation law: as you might have guessed from my mention of Maxwell's Equations, it's the law of conservation of electric charge.

This in itself doesn't seem earth-shaking, and it isn't--but in the middle of the twentieth century, when the number of subatomic particles kept growing and growing as higher energy experiments were done, and physicists were searching frantically for some way of bringing order to the chaos, they got a lot of mileage out of this idea of looking for symmetry groups. By that time, of course, quantum theory had taken over, and so the U(1) symmetry was not thought of as a symmetry of the electromagnetic field in the classical Maxwell sense, but as a symmetry of the photon, the particle of light. So the question became: can we find more complicated symmetry groups that will provide a structure to account for other known particles?

This line of thought turned out to be extremely fruitful, yet another example of an abstract mathematical structure turning out to have a deep physical significance. In the 1960's and 1970's it was shown that the weak force and the electromagnetic force were two aspects of the same force, the "electroweak" force, which had an SU(2)xU(1) symmetry group structure (the "x" here basically indicates that the SU(2) and U(1) groups are combined in the "obvious" way to form a larger group). Similarly, the strong nuclear force (the force that holds atomic nuclei together) was shown to have an SU(3) symmetry group structure. (The groups SU(2) and SU(3) are higher-dimensional "relatives" of U(1), and are also Lie groups--in fact, in some ways they relate to the quaternions and octonions, which I discussed above, in the same way as U(1) relates to the complex numbers. But that would be a much bigger digression.)

The current "standard model" of particle physics, which includes all three of the above forces (i.e., all forces except gravity), is thus based on equations having an SU(3)xSU(2)xU(1) symmetry group structure. One line of attack used by physicists who are currently searching for a "Theory of Everything" is to look at still higher-dimensional groups that have SU(3)xSU(2)xU(1) as a subgroup, and therefore might be candidates for the overall symmetry group of everything.

And now I really am out of breath.


If you want more information about Lie Groups and their relationship to physics (specifically quantum mechanics--but really that is their relationship to physics, since all of the correspondences I noted above between symmetry groups and conservation laws and so forth ultimately arise from quantum mechanics), there are two good articles on John Baez' web site (which is also one of the sites that mirrors the Usenet Physics FAQ): one on Lie Groups and Quantum Mechanics, and one on Elementary Particles and the symmetry groups underlying them.

Also useful is the MathWorld Entry on Lie Groups. It has links to a lot of other MathWorld articles on related concepts, which are useful in getting your bearings if you're not used to the subject.

Also, a note about the quaternions and octonions and vector spaces. I said that while it is possible to construct general vector spaces in any number of dimensions, it is *not* possible to construct hypercomplex division algebras in any number of dimensions. I didn't want to go into details here (since I don't fully understand them), but the basic reason can be seen from the following example. Suppose we try to construct a three-dimensional hypercomplex division algebra; in other words, instead of just one imaginary unit (square root of -1) we try to have two (i and j). The two of them will have to anticommute (that is, ij = -ji) for things to work out right, and given that, if you work out the square of ij (call this k), you will find that it is also -1! That is, you can't have a hypercomplex division algebra with just two imaginary units--as soon as you have two, you automatically get a third.

(Why do i and j have to anticommute? Because if they commute then they are really the same; work it out and you'll see that j has to be equal to either i or -i if i and j commute, so j isn't really a different basis vector unless it anticommutes with i.)