TITLE: When $G/H$ is cyclic?
QUESTION [6 upvotes]: Clearly $G/H$ is cyclic if $G$ is cyclic. But sometimes $G/H$ is cyclic without $G$ necessarily cyclic, for example: $(\mathbb{Z}_6\times \mathbb{Z}_6)/\left \langle (2,3) \right \rangle \cong \mathbb{Z}_6$
In cases like the latter, how can one determine if the group is cyclic? I mean, is there something in general that lets me know?
For example

Write $(\mathbb{Z}_{20}\times \mathbb{Z}_6)/\left \langle (10,2) \right \rangle $ as an external direct product of cyclic groups of prime power order.

Is $(\mathbb{Z}_{20}\times \mathbb{Z}_6)/\left \langle (10,2) \right \rangle $ isomorphic to $\mathbb{Z}_{4}\times\mathbb{Z}_{5}$ or isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_2\times \mathbb{Z}_5$?

REPLY [3 votes]: If $|G/H|$ is prime,  then of course it's cyclic.
In general,  the Smith normal form can be used to compute the structure of quotients of $\mathbb Z^n $.  But every finitely generated abelian group is such a quotient. After determining $G/H$ as a quotient of the free abelian group on  $n$ generators, this gives an approach.