Halp

Does there exist a finite-index subgroup of the real numbers under addition? O_o

Hmmm, I'd say almost certainly yes.
If so, have Such a subgroup H in R.

Then R/H is a group of order n.

consider rH (coset), H=(rH)^n=r^nH and so r^n=nr is in H.

but notice that for all r, r/n is in R, and so by the previous argument n(r/n)=r is in H.

But than R is a subgroup of H. CONTRADICTION.

God I'm good.