Question

Show that q, the group of rational numbers under addition, has no proper subgroup of finite index

Answer 1

I assume that the subgroup in question is non-trivial, otherwise we have {0}.

Suppose that Q is the group of rational numbers under addition.

Now suppose that P is a finite indexed non-trivial subgroup of Q.

Then, there exists g ∈ P such that g ≠ 0.

This means that we also have ng ∈ P  ∀n ∈ Z (the integers).

Thus we are forced to conclude that P contains infinitely many elements.

                                                                                                                      Q.E.D