Another nice arithmetic application of (cyclic) group theory is the fact that the multiplicative group of a finite field is cyclic, or (in down-to-earth terms) that one can obtain every non-zero residue class modulo a prime just taking consecutive powers of a single well-chosen one.
I always give this example in my undergraduate Algebra course because it also gives me the occasion to show that in maths there are still questions which are unanswered but can be easily understood by a beginner (obviously, I'm referring to Artin's conjecture in this context).
