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

Question

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?

Thanks for your time.

Answer 1

Yes, it's fine. You have been formal and you also correctly cited the needed isomorphism theorem!