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.