I am struggling with the following question.
Suppose I have a group $H$ which is a subgroup of $\mathbb{Z}\oplus\mathbb{Z}$, such that any element $\begin{bmatrix} a \\[0.3em] b \end{bmatrix}$ is defined as: if $b=0$, then $a=0$. How can I prove that $H$ has a basis with exactly one element?