Class numbers! $SL_2(\mathbb{Z})$ acts on integral binary quadratic forms, probably stick to the negative discriminant (positive definite) case. It is not too hard to write out a proof of a fundamental domain for the action of $SL_2(\mathbb{Z})$ (at least if the student has some proof to read, and then is expected to write it in his or her own words), from which the finiteness of the class number is immediate.

There's no reason that the students need to know the abstract definition of a group action here as everything is quite explicit.

The fun part is that you can then compute class numbers to your heart's content. For example, it is easy to compute that $h(-163) = 1$. Does that ever happen again? Nope! Gauss couldn't prove it. But the student who computes $h(-163)$ (and maybe $h(-167)$ and $h(-171)$, etc.) by hand is likely to appreciate why Gauss believed it.