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What are some interesting applications of the concept of homomorphism?

Example: If there is a homorphism from a ring $R$ to a ring $r$ then a solution to a polynomial equation in $R$ gives rise to a solution in $r$. e.g. if $f:R \rightarrow r$ and $X^2+Y^2=0$ then $f(X^2+Y^2)=f(0), f(X^2)+f(Y^2)=0, f(X)^2+f(Y)^2=0, x^2+y^2=0$

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meta discussion here:… – Tom Boardman Sep 3 '10 at 8:55
@user1613: I would appreciate if you could be much more specific about what you want. Are you a beginning student of abstract algebra who wants to understand why people care about homomorphisms? Are you a first-time teacher who wants good examples for his/her students? Etc. – Qiaochu Yuan Sep 3 '10 at 16:18
@user1613: In fact homomorphism is a widely concept even outside the mathematical subject we usually call abstract algebra. However, it is always a mapping between to structured objects of the same kind that and that map is structure preserving. By structure I mean an operation. – AD. Sep 3 '10 at 17:33
@Qiaochu Yuan: I studied some algebra years ago, mostly on the basis of memorizing stuff to pass exams. It seemed on the whole to be a lot of abstract nonsense. In particular I was nonplussed by homomorphisms. It wasn't a concept that felt nice and good. It gave more a feeling of discomfort. Then the other day I wrote the question about arithmetic of n-dimensional arrays and the above example about polynomial solutions occured to me and suddenly the concept of homomorphism felt good so I guess I'm looking for more examples that improve the taste of the concept of homomorphisms. more... – user1613 Sep 3 '10 at 20:00
@user1613: you should mention all this in the question. – Qiaochu Yuan Sep 3 '10 at 22:48

A professor of mine, Francisco Raggi, once said in class:

There are two ways to study a ring. You can stare at it, and stare at it, and stare a it, until you can spot interesting features of the ring, and then you can start describing those interesting features. Or you can study its images and the way it acts on other structures, by considering homomorphic images and modules over the ring. The latter is usually easier and more fruitful.

When you are studying a particular structure, you can study it in isolation: you can consider a vector space, and its subsets and subspaces, and say a lot of interesting things about that vector space; but in the end, it doesn't get you all that far. The same is true when you study a particular ring (and you can learn a lot just by staring at that particular ring; think of classical Number Theory as the result of staring intently at the integers looking for interesting features), or groups, or topological spaces, etc. But it turns out that studying the functions between similar objects that preserve the structure (linear transformations for vector spaces, ring and group homomorphisms, continuous maps, etc) leads to a much richer palette, with a lot more information and a lot more structure to work with. In some cases, like vector spaces, the new objects have themselves the same structure that you were studying (the set of all linear transformations $T\colon\mathbf{V}\to\mathbf{W}$ can be given a natural structure of a vector space, so everything you know about vector spaces applies) or some other structure (if $A$ and $B$ are abelian groups, then the set of all group homomorphism $A\to B$ can be given a ring structure by pointwise addition and composition, so you can study it as you would study a ring) which in turn gives you a wealth of information about the structures themselves. And of course, things like the Isomorphism Theorems can give you a lot of information about the structure you are looking at by looking at its images, which will often be "simpler" or "smaller" than the one you were originally concerned with.

Even in the case were we have found a lot of good information by simply staring (as in classical number theory), just look at how big a leap forward was achieved with the introduction of congruences (homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}/m\mathbb{Z}$), or with algebraic number theory (mapping to larger rings and then trying to import the information back into $\mathbb{Z}$).

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Here are some striking examples from looking at the Lebesgue space $L^1(\mathbb{R})$ under Fourier transform. $L^1(\mathbb{R})$ is an algebra under convolution, that is it is a vector space with a multiplication (here multiplication is the convolution). The Fourier transform $\mathcal{F}: L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an algebra homomorphism from $L^1$ into a sub-algebra $\mathcal(L^1(\mathbb{R}))$ of the space $C_0$ (continuous functions that vanish at $\pm\infty$).

  1. In particular $$\mathcal{F}(f*g) = \mathcal{F}(f)\cdot\mathcal{F}(g)$$ and instead of studying convolution equations in $L^1$ we may study multiplicative equations in the Fourier image.

  2. An other example is The Wiener Tauberian Theorem, that states that the translates of $f$ span a dense subspace of $L^1$ if and only if $\mathcal{F}(f)$ is non-zero.

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Since negation qualifies as a homomorphism between conjunction on {0, 1} and disjunction on {0, 1}, I'd expect (I'm not an engineer) that all over digital electronics you have applications of the homomorphism concept (this isn't to say that the engineers realize it). Possibly your computer applies this concept, in some sense, several times in one way another while you read this text. Conversion between conjunctive normal form and disjunctive normal form (DNF) often uses the De Morgan laws, which can get said to qualify as an application of the concept. I'd think that Karnaugh maps, the Quine-McKluskey method, finding prime implicants, etc. for finding minimal DNFs can get said to implicitly use the concept of homomorphism, since they may get said to implicitly use the De Morgan laws, as I recall.

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