5
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.