How about the result concerning finding the roots of polynomials; that there are increasingly complex exact and explicit formulae as you go from linear to quartic, but it's impossible to find such formulae for quintics or higher.

I don't know if that fits your needs: it's elementary to state, but not exactly elementary to prove.

Lovely result, but as you expect, not really sufficiently elementary. Also, while mathematicians find impossibility proofs among the most useful results, non-mathematicians are usually more impressed by results that show something unexpected is possible.

But I'll keep it in mind.

Something slightly interesting might be the Rubik's cube group. By considering the Rubik's cube a group under the rotation operation, then we can essentially completely solve the Rubik's cube puzzle.

If you are familiar with the Quaternions, take a look at the 3D rotation group SO(3). This group apparently has applications in computing where one needs a way to perform rotations in three dimensions. According to the Wikipedia article, https://secure.wikimedia.org/wikipedia/en/wiki/Quaternions_a..., this method of representation has certain important advantages over other methods, including being numerically stable and avoiding a problem called gimbal lock, where rotation about one of the possible axes is no longer possible.

Well, the idea of using commutators in the Rubik cube is what leads to a nice, elementary method of solving it, but it's not really a theorem or result from group theory that's telling us something we didn't already know. Plus I suspect many people would regard this as just a toy, a form of "play for play's sake" and hence still irrelevant.

The fact that the quaternions form a field (hence two kinds of group, one of which is interesting) where the multiplication doesn't commute, and then that can be used to model rotations is again interesting, and can be asserted to be useful (and it is) but it's still not really what I'm looking for.

But keep the ideas coming. There's got to be a good one out there somewhere - isn't there?

I'd submit Euler's theorem, http://en.wikipedia.org/wiki/Euler%27s_theorem. This result is the heart of the Rabin-Miller primality test, which is what makes finding large (probable) primes easy, making RSA encryption practical.

Hmm. I do use that result in the context of number theory, and I suppose I could bend it to use it as an example of groups in abstract being used in practice.

I'll have to think about that, but it does show promise. The question is, how can I present it as a result that's specifically from group theory being used in number theory? Does that make sense? I'd like to be able to demonstrate something that's specifically group theory, and I'm finding it hard to extract the abstract, in this case.

But it shows promise.

What about (not) Burnside's lemma:

If $G$ acts on a set $X$ and $|X^g|$ denotes the number of elements of $X$ fixed by $g$, then the number of orbits $|X/G|$ is given by

$|X/G| = \frac {1} {|G|} \sum_{g \in G} |X^g|.$

The proof is simple, yet you can count an amazing number of things with it.

Ah, now that's more interesting. I'm not sure I can make that readily accessible, but if I consider a specific example (yet to be invented) then maybe I can make it "obviously useful" while still being an abstract result.

I'll have to think about that. Boffo. Good one. Thanks.