Show the quotient groups of two nonzero rational numbers are isomorphic.

Question

Let x, y be rational numbers. Show Q/$\langle x \rangle$ is isomorphic to Q/$\langle y \rangle$.

I know to show isomorphic you need to show that there is a bijection and f(uv) = f(u)*f(v), but I can't think of such a bijection

Answer 1

Hint: Try multiplication by $\frac{x}{y}$ as a map $\mathbb Q/\langle y\rangle \to \mathbb Q/\langle x\rangle$. You'll need to show that this gives you a well defined map, that this map is a homomorphism, and that this homomorphism is bijective.