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I'm going to be a teaching assistant for an undergraduate class in abstract algebra next semester, for students who have not taken abstract algebra before. It will deal with group theory and linear algebra, but the students have already had a semester of linear algebra so I'm thinking about how to deal with group theory.

I'd like to present some examples of applications of group theory that motivate the theoretical questions the course will deal with. For instance, "symmetries of three-dimensional objects form groups. Crystals have symmetrical structure. If we could get some bounds on possible forms of three-dimensional symmetry, this would give limits on the sorts of crystals that could form, which would be interesting to a chemist." Another example could be, "The roots of a polynomial can sometimes be permuted in ways that do not change the value of polynomial functions of the roots, and these permutations form a group. [insert explanation of Galois theory here] If we could determine whether this group is trivial, we could see whether solving the polynomial is possible."

My goal is to persuade the students that group theory is useful and therefore interesting, but since they will most likely be interested in different things, I'd like to have a big list in the hope of being able to find something for each of them. Can people suggest either applications of group theory or places to find such applications?

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How about Quantum Mechanics? – Arturo Magidin May 23 '10 at 2:50
See also… – Gjergji Zaimi May 23 '10 at 2:57
If you are going to post anything on the web, don't neglect the visual part (crystals make for great illustrations!). There are very nice cartoons in "Indra's Pearls" that lighten up abstract theory, but they are copyrighted, obviously. – Victor Protsak May 23 '10 at 7:47
Your comment about seeing "whether this group is trivial" for Galois theory is very misleading. Nobody discovers that a polynomial splits over the rationals by computing that its Galois group is trivial. – KConrad Feb 15 '11 at 15:21

I had the same question before I taught a course that was largely group theory. Here is the webpage I created to address the issue:

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That's a nice motivational page, Keith! Personal anecdote: before going to Max-Planck-Institut in Bonn for a year, I was advised to come up with good one-sentence German answers to basic questions and possible follow-ups, like "What is your profession": "I am a mathematician", "Ach, so... what do you actually study?": "Representation theory". I've decided that if they want to know what does $\textit{that}$ mean, I'd reply that it's a study of fundamental symmetry (I've translated Q&A into English for convenience). Actually, it did come in handy more times than I would have expected. – Victor Protsak May 23 '10 at 7:38

In case some of your students like physics or engineering:

One of the last chapters of James and Liebeck's textbook on Representations and Characters of Finite Groups explains how group theory (especially some basic character theory) can be used to radically simplify calculations in a type of physics problem: the vibration of a molecule. You might be interested in this as it takes a big linear algebra problem (say 15 x 15, not so large it cannot be written down on paper) and makes it a much smaller problem (say 3 x 3, something they might be more inclined to solve). We regularly assign problems that look like the first versions of these in calc 2, and some of the medium sized ones in calc 4, so perhaps they may help encourage continuity between courses.

We studied steady states of "trusses" and other aspects of bridge construction in our second undergrad linear algebra class, and I found it fascinating. I would have been very impressed that group theory could have dramatically shrunk those problems.

There are some undergraduate texts on the hydrogen atom that are basically representations of groups. Several good texts are available online from Springer, and can be used to fill out Arturo's suggestion of quantum mechanics. Again these will mesh very well with linear algebra, especially a second course.

If the students are familiar with differential equations, then exploiting simple symmetries in the equations to find more solutions is good. The clearest example of this is a DE with real equations and a complex solution; the equation is invariant under complex conjugation, so the complex conjugate of the solution will also be a solution. One can use larger symmetries, such as a rotation invariant PDE etc.

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An obvious one is proof of Fermat's little theorem from Lagrange's theorem. This has immediate application in public-key cryptography, which computer people will care about.

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none -- good point but just for the record: I think it's Euler's theorem, rather than Fermat's, that's used in public key cryptography (RSA). – algori Feb 15 '11 at 18:05
Also, this was known long before real group theory was developed. The proof was implicitly group theoretic. You can understand it in every detail without knowing any group theory at all. So Lagrange is Overkill here. – Martin Brandenburg Aug 17 '11 at 18:52
@algori: It is written in many places that Euler's theorem is used for RSA, but for the record it is not true. To show RSA works for all messages, not just those that are relatively prime to the modulus $pq$, Fermat's little theorem suffices. Look at the proof of correctness of the RSA algorithm at the end of the Wikipedia page on RSA. Using Euler's theorem instead of Fermat in that proof leaves uncertainty about messages having a factor in common with the modulus. Even though that is a very unlikely event, it's nice to know RSA would work even for those and the proof uses Fermat. – KConrad Oct 20 '13 at 16:35

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