Find a cyclic subgroup of $\mathbb{Z}_{40}\oplus \mathbb{Z}_{30}$ of order $12$ and a non-cyclic subgroup of $\mathbb{Z}_{40}\oplus \mathbb{Z}_{30}$ of order $12$.
I am having a lot of trouble understanding how to do this problem. If someone could please explain step-by-step I would appreciate it.
For the first one, consider the subgroup generated by $(10,10) \in \mathbb{Z}/40 \oplus \mathbb{Z}/30$. (This is a copy of $\mathbb{Z}/4 \oplus \mathbb{Z}/3$ inside $\mathbb{Z}/40 \oplus \mathbb{Z}/30$, and this is cyclic because $3$ and $4$ are coprime.)
For the second one, consider the subgroup generated by $(20,0)$ and $(0,5)$. (This is a copy of $\mathbb{Z}/2 \oplus \mathbb{Z}/6$ inside $\mathbb{Z}/40 \oplus \mathbb{Z}/30$, and this is not cyclic because $2$ and $6$ are not coprime.)