It's time to live up to the description of this blog: "an exploration of structure in things."

What is structure? Structure can broadly be defined as symmetry; the commutative property is an example of a symmetry that gives some operation structure. Today I'd like to talk about the kinds of structure that we can find in a set and a binary operation on it and how to translate that structure to other sets and other operations.

Let's talk about addition for now; in particular, I'd like to talk about the addition of real numbers . Addition has the following structure:

(commutativity)
(associativity)
(identity)
(inverses).

Together, these four types of structure define a (commutative, or abelian) group. (A group is not necessarily commutative). There are plenty of good places to talk about groups in general; here we will primarily be concerned with structure-preserving maps, or homomorphisms, between groups (and other things). I'd also like to restrict this discussion to things that you may have already seen (but not in a group-theoretic context) rather than introducing either new theoretical examples or talking a lot about geometry (yet).

A homomorphism of groups translates the group operation on to the group operation on . In other words, . The on the LHS denotes the group operation on while the on the RHS denotes the group operation on ; note that these do not have to be the same!

In my experience, dense definitions do not necessarily communicate important concepts. Therefore, let me give an example of a useful group homomorphism. Multiplication of (positive) real numbers has the following structure (again, by itself):




.

Note that we needed to specify the positive reals in order to have inverses. You might now be able to predict the next step. The function



(for any real ) is a group homomorphism that takes addition of real numbers to multiplication of positive reals! In fact, if this homomorphism is invertible:



is a group homomorphism that takes multiplication of positive reals to addition of reals. An invertible homomorphism is called an isomorphism. Groups that are isomorphic are, for all practical purposes, "the same." Any given calculation, say



in one group can be translated into the corresponding calculation



in the other. Furthermore, in this case the only differentiable isomorphisms between the addition of reals and the multiplication of positive reals are functions of the above form where can be complex.

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When more interesting things happen. The map no longer maps to the positive reals, and it is no longer invertible: rather, it maps to the circle group (under multiplication) and takes the value (the identity) whenever .

This set is called the kernel of a homomorphism. Note that if the kernel is not trivial, then the homomorphism is not invertible: if , then




so must be in the kernel of .

"But," you say, "we can repair injectivity if we take to be the set of angles under addition !" This is very true; in fact, this intuitive notion of "modding out" by the kernel leads to the first isomorphism theorem. We'll have more to say about kernels once we talk about things other than groups.

For a generalization of the exponential map, see matrix exponential.

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In additive notation, the definition of a homomorphism can be written . You might recognize this as the Cauchy functional equation. The "normal" solutions are the functions . When , these are isomorphisms from the addition of the real numbers to the addition of the real numbers. An isomorphism from a group to itself is called an automorphism, and can be regarded as another kind of symmetry.

The fascinating thing about the Cauchy functional equation is that it has bizarre solutions that are not of the form ; these bizarre solutions are dense in the plane and continuous and differentiable nowhere. The implication that can be drawn here is that the addition of real numbers is extremely complicated (relative to other groups) because it has a large set of nontrivial automorphisms.

In multiplicative notion, the definition of a homomorphism can be written . We've talked before about fully multiplicative functions before, the Dirichlet characters, which are homomorphisms from the multiplicative group to (a subgroup of) the circle group. If a primitive root exists , the subgroup of the circle group we get is cyclic and is generated by one element (a primitive root of unity). See the connection?

If we write and write multiplicatively and additively (get used to both conventions), the definition of a homomorphism can be written . We've already seen the logarithm when . Let's try instead and we will come to a rather fun conclusion.

Note that unique prime factorization extends to : we can write any rational number uniquely as where . We therefore have the following family of homomorphisms: define to be the exponent of in the prime factorization of (as we used here). Then we have the beautiful property that

.

is a homomorphism from to for any . Note that none of these homomorphisms are remotely invertible: the kernel of is the set of rationals with neither a numerator or denominator divisible by .

We know that a rational number is uniquely described by its prime factorization. Can we describe this in the language of isomorphisms? In fact, we can: the map



gives an isomorphism from to the group of integer sequences that are eventually zero with group operation componentwise addition. (That was a mouthful! Consider the analogy to an l^p space.) The fact that this map is an isomorphism is the content of the fundamental theorem of arithmetic. Thus one way to interpret the fundamental theorem of arithmetic is in terms of the properties of .

(An alternate definition of leads to the beautiful notion of a p-adic number. We may return to this subject later; we first need to switch from talking about groups to rings, and after that to metric spaces.)

Note that if there were a finite number of primes (say of them) then the above would be an isomorphism to . Thus, one way to interpret the result that there are an infinite number of primes is that is not finitely generated.

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I never finished my discussion from Rational solutions to Polynomials. The correct answer to the practice problems is the following

Theorem. If a cubic curve has repeated roots, then it is singular and isomorphic to the projective line. Otherwise, it is non-singular and not isomorphic to the projective line.

There is not enough space here to go into a thorough discussion of what happens in the non-singular case, which is a rich subject. I recommend Tate/Silverman; it's a fairly accessible introduction. What I can tell you is that (the rational / real / complex points on) a non-singular curve have a group structure, and if written in the Weierstrass form there exists a function such that



called a Weierstrass elliptic function, and moreover this function is a group homomorphism from to the points on an elliptic curve, where is a lattice spanned by two complex numbers called the fundamental periods of . For this reason, is called a doubly-periodic function. Note the analogy to the function and its derivative , which are defined on , parameterize circles, and have a single period. One way to think about the theory of elliptic functions is as a generalization of trigonometry to curves of degree greater than .

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Practice Problem 1: Find a group homomorphism from the circle group to some multiplicative group of matrices (without looking at the Wikipedia article!). Think rotation. Thus find a group homomorphism from the nonzero complex numbers to (a bigger) multiplicative group of matrices. (Note that matrix groups are not in general commutative.)

Practice Problem 2: is a ring, which has two related structured operations. We can define ring homomorphisms analogously to group homomorphisms: they must translate both the addition and the multiplication of some ring to some other ring. Verify that the "reduction modulo map" is a ring homomorphism. What is its kernel? (The kernel of a ring homomorphism maps to the additive identity.)