Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

So I'm trying to get this unmotivated student in my class interested in learning group theory. I've recently ascertained that he likes analysis a bit better than algebra, and that sequences and series are his favorite part.

Does anyone know of some cool sequences/series results which involve group theory in an essential way? I would be especially receptive to interesting problems with slick group theoretic solutions.

share|cite|improve this question
Periodicity of $a^n \pmod{p}$? – Calvin Lin Feb 20 '13 at 19:09
up vote 2 down vote accepted

Two things come to mind; one has to do specifically with groups and sequences, as you requested; the other is more generally about algebra and analysis.

The first is that the fractional parts of the powers of Pisot–Vijayaraghavan numbers tend to zero exponentially, which is quite surprising until one understands the algebraic background. One can gain a basic appreciation of this phenomenon without knowing much group theory, but the connections to Galois theory are evident enough to perhaps raise his interest.

The phenomenon that most convincingly demonstrated to me that algebra can sometimes be superior to analysis is the determination of the spectrum of the quantum harmonic oscillator. You can treat this entirely analytically, as a second-order differential equation to be solved, and you get wavefunctions given by a Gaussian multiplied by Hermite polynomials. The equidistance of the eigenvalues appears rather coincidental in this approach.

On the other hand, if you forget about the concrete wavefunctions and express the Hamiltonian in terms of ladder operators operating on abstract states, showing that the eigenvalues are equidistant becomes an almost trivial exercise, and you feel you've distilled the essence of the problem and all those fancy wavefunctions were just hiding it.

share|cite|improve this answer

Maybe Mal'cev-Neumann series? (see

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.