I would like help with the following simple question.
I have an abelian group given by the presentation $\langle a,b \mid ra = 0, ta = sb\rangle$.
The Smith Normal Form gives me that this group is isomorphic to $\mathbb{Z}_n\times \mathbb{Z}_m$, where $n = \gcd(r, t,s)$ and $m = rs/\gcd(r, t,s)$. Thus, it must be possible to express the group in the form $\langle c,d \mid nc = 0, md = 0\rangle$.
How can I express $c$ and $d$ in terms of $a$ and $b$?
Thank you for the answer.