I am not sure if the answers of the following question are correct, so if anyone could verify it for me, that would be grateful.
If I take $H:= \mathbb{Z} \times (2,2) + \mathbb{Z} \times(4,12)$, then I have to compute the following:
Compute the order of $(1,0) + H$ in $\mathbb{Z^2}/H$, I found that the order is 16 but I am not sure.
Calculate the rank and the elementary divisors of $\mathbb{Z^2}/H$ , I found that the elementary divisors are 2 and 8. The rank is 2.
For the next question I need help to solve it.
If we let $a,b,c,d \in \mathbb{Z}$ and take $H:= \mathbb{Z} \times (a,b) + \mathbb{Z} \times(c,d)$, then I have to show that: $\mathbb{Z^2}/H$ can be generated by a single element $\iff gcd(a,b,c,d) = 1$.