# Are $\Bbb Q/2 \Bbb Z$ and $\Bbb Q/5 \Bbb Z$ isomorphic as groups?

Are $$\Bbb Q/ 2 \Bbb Z$$ and $$\Bbb Q / 5 \Bbb Z$$ isomorphic as groups?

I take the map $$\pi : \Bbb Q \longrightarrow \Bbb Q/5 \Bbb Z$$ defined by $$a \longmapsto \frac {5} {2} a + 5 \Bbb Z,\ a \in \Bbb Q.$$ Then this map is clearly a surjective group homomorphism with kernel being $$2 \Bbb Z.$$ Hence by the first isomorphism theorem we have $$\Bbb Q / 2 \Bbb Z \cong \Bbb Q / 5 \Bbb Z.$$

Is my reasoning correct at all? Would anybody please verify it?