Develop Intuition with Abstract Algebra: Groups & Rings

[mrb]

I've taken 2 (undergrad) courses in abstract algebra and a reading course in Galois Theory, and I still don't understand the point of studying groups and rings. The courses have not been particularly difficult for me, but my motivation is extremely low.

In Galois Theory obviously I saw an application of groups to solve a problem. And I can follow the proofs. But I still don't understand the connection between groups and solvability by radicals in any intuitive sense. And I haven't gained any insight into how to apply groups to solve other problems. This is completely different from analysis, where it is extremely clear to me how it can be used to solve various problems. And also in analysis, I have enough intuition and insight about it that I can prove theorems on my own readily.

How do people develop intuition with groups and rings? Can you develop a visual or geometric or physical sense about them, or is it forever just symbols on a page that you manipulate by the rules? Do people actually enjoy this? What other problems can groups and rings solve?

Thanks.

 

[n!kofeyn]

I myself don't know a whole lot of algebra, but it is very important in many different areas of mathematics and physics. The best example I can think of is topology. A big area of topology is determining whether two spaces are homeomorphic or not. This just means can we bend, stretch, or compress without ripping the shapes into the same thing. This can be difficult to show, but you can introduce algebra into topology. There are some things called homology groups where you can determine if two topological spaces are the same if their homology groups are isomorphic (the algebraic version of homeomorphic). That's all I really know, but since you already know some algebra, I would recommend reading through an earlier edition (like the 4th) of A First Course in Abstract Algebra by John Fraleigh. He does a good job motivating things and has a section on homology groups. Other than that, maybe pick up an algebraic topology book.

Also, there is calculus on manifolds. It is a generalization of calculus to manifolds, which are abstractions of Euclidean space. I just learned this topic this past semester, but there is a book called Smooth Manifolds and Observables by Jet Nestruev that basically expalins how differential calculus is a natural part of commutative algebra. Now again, I don't know enough algebra to go any further, but I thought I would mention it to let you know of the book, which is probably advanced though, and the usefulness of algebra.

 

[morphism]

First a minor correction to n!kofeyn's post: There are non-homeomorphic spaces with isomorphic homology groups. It works the other way around, i.e. if two spaces are homeomorphic (or even homotopically equivalent) then their homology groups are isomorphic.

Groups, rings, etc. are just abstractions of things a mathematician deals with every day. For example, commutative rings can be thought of as a generalization of the set \mathbb{Z} of integers with its additive and multiplicative structure. One additional thing we have in \mathbb{Z} is unique factorization into primes; this is such a deep property that it's profitable to see how it generalizes to arbitrary commutative rings, where of course we would need an appropriate analogue of "prime numbers". (This was an important observation back when people were trying to prove Fermat's Last Theorem. Actually, a lot of classical algebraic notions owe their existence to FLT.) Well it turns out that many rings, even nice ones, don't have any sensible unique factorization into primes. From an advanced point of view, the ring \mathbb{Z} is special because its ideal class group is trivial. What does this mean? To each "nice" ring R (by which I mean at least a Dedekind domain) one can associate (by following some sort of procedure) a special group G(R) which measures the extent to which unique factorization fails in R. For example, it turns out that G(R) is the trivial group if and only if R is a unique factorization domain. The ideal class group is a basic object of algebraic number theory.

This idea of associating algebraic invariants to things is very common in modern mathematics. Classically, mathematicians would associate numerical invariants to objects in order to distinguish between them. Here's a stupid example: we know that a square is different from a triangle because they have a different number of sides. A less stupid example would be to note that the topological space (0,1) \cup (2,3) \cup (4,5) is not the same as (=homeomorphic to) the space (0,2) \cup (5,6) because the former has 3 connected components wheras the latter has only 2.

Here's a more interesting example. To distinguish between compact Riemann surfaces, 19th century topologists associated to each such surface an integer called its genus, which basically counts the number of holes in the surface. Thus the sphere has genus 0, a torus genus 1, a 2-holed torus genus 2 and so on. And it turns out that a compact Riemann surface is completely determined by its genus. More generally, we can distinguish between closed surfaces (=closed connected 2-dimensional manifolds) by looking at two numerical invariants: their orientability (say, +1 if orientable and -1 otherwise) and something called their Euler characteristic (an integer). More generally still, we can use Betti numbers to try to distinguish between spaces. The Betti number of a space is roughly the maximum number of times we can cut the space before splitting it into two disconnected pieces.

There are plenty of other numerical invariants we can attach to a topological space, and 19th century topologists were almost exclusively concerned with coming up with new ones. But then came along Poincaré with the inspired idea of attaching an algebraic invariant to a topological space. He essentially defined the fundamental group (a.k.a. Poincaré group, or first homotopy group) of a topological space. What this group does is count the number of holes in the space (yes, this again! And this is not the end of it. There are several invariants concerned with counting holes in spaces. The reason for this is because this is actually a difficult thing to do in general) by looking at how loops can be continuously shrunk to a point inside the space. For instance, every loop in the plane can be continuously shrunk to a point, whereas in the punctured plane the loops around the puncture point cannot be shrunk without leaving the space. This information is encoded in the fact that the fundamental group of the plane is trivial, whereas the fundamental group of the punctured plane is not (it is isomorphic to \mathbb{Z}).

And n!kofeyn already mentioned the homology groups. These things count the number of holes as well, but in a more sophisticated way. There are also cohomology groups, K-groups, etc. etc. Some of these things are not only groups, but also rings, modules, vector spaces, etc. (And not all of them count holes!)

OK, enough about algebraic topology. Since you mentioned analysis then let me give you an example of where groups show up there. First let me say there are plenty (and I mean plenty) of such examples. There's no reason why I've chosen this one. Presumably you've done some or at least heard of Fourier analysis. There one begins by studying periodic functions on a compact interval of the form [0,2pi]. What's really going on, though, is that you're studying functions on the unit circle. The unit circle is a group (think of it as the set \{e^{ix} : x \in \mathbb{R}\} with complex multiplication), and Fourier analysis takes full advantage of this fact. Indeed, the unit circle S^1 is an example of what's called a topological group, which means a group with a topology which respects its group structure (i.e. the group operations of inversion and multiplication are continuous); in fact the topology is locally compact and the group is abelian, making S^1 into something called a locally compact abelian (lca) group. The Fourier transform, for instance, is really just a special map from the space L^1(S^1) of integrable functions on S^1 into the space c_0(\mathbb{Z}) of sequences which are indexed by the integers and which converge to 0 at both ends (these are none other than the Fourier coefficients of the L^1(S^1) function). It is possible to do Fourier analysis on other lca groups as well, and lots of nice things generalize (e.g. Poisson summation, the Plancherel formula, Fourier inversion, etc.). This is called commutative harmonic analysis. In fact we can also attempt to generalize Fourier analysis to nonabelian groups; this is the subject of noncommutative harmonic analysis. And harmonic analysis has profound connections to Lie theory, another beautiful area of math where groups are centre-stage.

And I could go on and on and on. Groups, rings, etc. are really every day objects for the mathematician, and this is why they're ubiquitous. They also form the foundation upon which the whole of mathematics rests.

Hopefully my ramblings here are useful (and also mistake- and typo-free!).

 

[n!kofeyn]

First a minor correction to n!kofeyn's post: There are non-homeomorphic spaces with isomorphic homology groups. It works the other way around, i.e. if two spaces are homeomorphic (or even homotopically equivalent) then their homology groups are isomorphic.

Thank you for the clarification morphism.

 

[mrb]

Thanks for the replies.

n!kofeyn - The book you suggested looks like exactly the kind of thing I want to see. Unfortunately it will be a while before I have a chance to look at it.

morphism - Thank you for your explanation of algebraic topology. I am aware of the general ideas, although I haven't worked with them. Same thing with Fourier Analysis. But none of this is concrete for me right now. Maybe after I actually work with these concepts I will see the light.

The thing is... a student's first exposure to analysis is a basic calculus class. And there (at least for me), right from the beginning, there are connections... to geometry, to physics, etc. These helped me understand why people care about analysis, and helped me come to appreciate it on its own terms, without needing to see outside applications.

In contrast, with algebra... the first thing in the class was the definition of a group, with no motivation. To me this is like if the first thing you got in a calculus class was the epsilon-delta definition of a limit, with no motivation. Is this really the way it's done? You are supposed to see a year's worth of theory as an undergrad, and only see it put to any use in grad school?

 

[n!kofeyn]

In contrast, with algebra... the first thing in the class was the definition of a group, with no motivation. To me this is like if the first thing you got in a calculus class was the epsilon-delta definition of a limit, with no motivation. Is this really the way it's done? You are supposed to see a year's worth of theory as an undergrad, and only see it put to any use in grad school?

I agree that there are problems with the teaching of abstract algebra. I myself was turned off of it because there was no intuition building of it, and this is why I have forgotten almost all of the algebra I once knew. I took a course in undergraduate that used the book Applied Abstract Algebra by Lidl and Pilz. I can't say much about the book as it has been 2 or 3 years since I took the course, but it has applications of algebra. In particular, it discusses coding theory, cryptology, image understanding, and fast Fourier transforms, among others. It is worth checking out as well as the above mentioned books.

 

[mordechai9]

I took only one semester of abstract algebra and I felt pretty much the same way as the OP. Actually, I probably felt even worse, because we didn't get into Galois theory at all. The most I understood about the material was "Yippee, this characterizes the rotations of a triangle/square/blah!"

Partly also, I felt really uncomfortable with the material, because it seemed much more abstract to me than analysis. I suspect this was partly because I had limited background in algebra, and future classes would probably be much easier for me/us. However, I also have come to understand that there is a broad class of mathematicians who love analysis, and basically hate algebra, or vice/versa, so I don't know, maybe this is a weird predisposition thing.

Morphism's post is outstanding... that's one of the best casual descriptions of the motivation/applicability for algebra that I've read, actually, I wish I could read a lot more along that same vein! I'm sure there must be books out there that focus on this applied/interrelated aspects of algebra. However, I have to admit, in the last lecture of my undergrad algebra course, we sort of had an open discussion, and I asked the very distinguished prof. "How does algebra relate to analysis?" and his response was literally along the lines of "Well, not at all really." ...

 

[morphism]

In contrast, with algebra... the first thing in the class was the definition of a group, with no motivation. To me this is like if the first thing you got in a calculus class was the epsilon-delta definition of a limit, with no motivation. Is this really the way it's done? You are supposed to see a year's worth of theory as an undergrad, and only see it put to any use in grad school?

A good instructor would no doubt motivate the definition of a group.

One can do this for example by using the notion of symmetry in plane geometry (think about the dihedral group) or by looking at permutations of a fixed set S (these things form a group under composition) or by studying the integers mod n first (lots of nice, concrete things can be said about Z_n!). The concept of a group can then be introduced as an abstraction of these ideas.

You can also see applications of group theory early on. Didn't your course go over, say, Burnside's lemma or group actions?

 

[mordechai9]

A good instructor would no doubt motivate the definition of a group.

One can do this for example by using the notion of symmetry in plane geometry (think about the dihedral group) or by looking at permutations of a fixed set S (these things form a group under composition) or by studying the integers mod n first (lots of nice, concrete things can be said about Z_n!). The concept of a group can then be introduced as an abstraction of these ideas.

You can also see applications of group theory early on. Didn't your course go over, say, Burnside's lemma or group actions?

In my class, (not speaking for the OP), my instructor actually motivated things this way, starting out with the permutation group and the dihedral group, etc etc.

Morphism - could you comment more about the Fourier transform thing you mentioned before? Is it right to assume that all integral transforms (Laplace transforms, etc.) work out because they define isomorphisms between groups? Why do these groups have the intrinsic property that higher frequency (high mode number, integer number, however you want to call it) components decay to zero? I mean, how is this property embedded into the group structure?

 

[mrb]

A good instructor would no doubt motivate the definition of a group.

One can do this for example by using the notion of symmetry in plane geometry (think about the dihedral group) or by looking at permutations of a fixed set S (these things form a group under composition) or by studying the integers mod n first (lots of nice, concrete things can be said about Z_n!). The concept of a group can then be introduced as an abstraction of these ideas.

Actually I exaggerated: he did talk about the dihedral groups as symmetries of regular polygons before he introduced groups in general. That didn't help me much, though, because it still wasn't clear *why anyone cared*.

You can also see applications of group theory early on. Didn't your course go over, say, Burnside's lemma or group actions?

No.

Last night I was flipping through a geometry book I never finished; it had a chapter on geometric transformations. It discussed Klein's Erlangen program of defining geometries by different transformation groups which kept different things invariant. This really gave me a little insight into what groups are good for.

 

[mal4mac]

Sounds like you should be taking more physics classes! If you need all this 'real world' motivation then you shouldn't purely be doing pure mathematics. I did group theory in a pure mathematics department and enjoyed it just as a pure mathematical game.

I mostly did courses in physics and had some idea that groups were applied in physics somewhere, but I never got to actually apply group theory to physics, only those doing single honours physics got that far. So if you want to see group theory applied to the real world take a physics degree!

 

[mrb]

Sounds like you should be taking more physics classes! If you need all this 'real world' motivation then you shouldn't purely be doing pure mathematics.

Actually, yes I do wish I had done more physics. All I have had is the mechanics class, and I will be taking E&M this fall (my last semester). I almost have a CS minor (there were scheduling issues with the last class I needed and I decided it wasn't worth the hassle); I wish I had done a Physics minor instead.

However, my issue is not that I really have to see "real world" applications, although that's fine. What I wanted was just to see some strong connections to SOMETHING. For instance, if our class had been shown how algebra can be used to study number theory, that would have been great.

GH Hardy talks about the difference between studying real math and studying chess in A Mathematician's Apology. I don't have the book with me, but he says something along the lines of: you're not going to get any deep insight about anything or make any great connections by studying chess, even though of course it is perfectly possible to make a mathematical theory of it and prove theorems and whatnot. For me, studying groups seemed like studying chess.

 

[mrb]

For anyone else with the same question I had:

I'm taking a grad course in commutative rings this fall. The course text is Kaplansky's Commutative Rings, but I bought two books by Miles Reid as supplements: Undergraduate Commutative Algebra and Undergraduate Algebraic Geometry.

Just skimming the Reid books has opened my eyes a little about what algebra is good for and has actually made me excited about the course. It's such a contrast to the Kaplansky book... Kaplansky has a very clear writing style, but it seems to be the kind of book where you could work through the whole thing and follow every proof and come out of it having no idea why you should have cared.

Also, Reid includes little essays at the end of his books complaining about basically the same thing I complained about, which was refreshing. He says teaching algebra from the purely abstract approach, separated from motivation or applications, is a big mistake.

 

[morphism]

Morphism - could you comment more about the Fourier transform thing you mentioned before? Is it right to assume that all integral transforms (Laplace transforms, etc.) work out because they define isomorphisms between groups? Why do these groups have the intrinsic property that higher frequency (high mode number, integer number, however you want to call it) components decay to zero? I mean, how is this property embedded into the group structure?

I'm afraid I can't say anything intelligent about these things. What the Fourier transform means to me is very different from what it means to a physicist.

Can you give me an example of a "higher frequency component" and explain why it decays to zero?

 

[mordechai9]

Ah, well I thought that the first two questions were mathematical questions anyways...? But I'm guessing you just don't know any more about that.

Regarding the "damping of high frequency components", I meant -- If you have a signal with sharp peaks in it, the Fourier transform of this signal will incorporate more high-frequency components. This is clear just because the Fourier transform is composed of sine and cosine functions which are very not-sharp, so you have to take more and more of them (i.e., add in higher frequency components) in order to approximate the sharp corners. But in physics, basically sharp signals represent anomalous events, for example, shock waves, which are damped out and decay very quickly. As another example, in quantum mechanics, the probability distribution is usually considered a smooth, differentiable function without any sharp peaks. In general in physics, there is an overall trend where going higher into the frequency spectrum you expect lower amplitudes. This can also be interpreted by recognizing that the higher frequency components have higher energies, and of course, energy is a conserved commodity. Sorry, kind of long-winded, but that's what I was referencing.

Edit: Note that purely mathematically also, of course, the high frequency components of the Fourier series must go to zero for the series to converge.