Is sublattice of $Z^2$ generated by at most 2 elements?

Question

Is it true that sublattice of $Z^2$ is generated by at most 2 elements?

By $Z^2$ I mean the group of pairs of integers with addition coordinatewise.

Answer 1

Yes. Let $G$ be a subgroup of $\Bbb Z^2$. In general the following procedure gives two generators for $G$. First take $(a,b)\in G$ with $a>0$ minimum possible, and then take $(0,c)\in G$ with $c>0$ minimum possible. Then $G$ is generated by $(a,b)$ and $(0,c)$.

There are some subgroups where this fails: either every element in $G$ has first coordinate zero, or there are no nonzero $(0,c)$ in $G$. But in these cases, $G$ only needs at most one generator.

More generally, each subgroup of $\Bbb Z^n$ requires at most $n$ generators.

Answer 2

Yes. Z^2 is the fundamental group of the torus $S^1\times S^1$. By covering space theory, any subgroup is the fundamental group of a covering space of the torus. But the only covering spaces are topologically $S^1\times S^1, S^1\times\mathbb{R}, \mathbb{R}\times\mathbb{R}$, and all of these have fundamental groups with at most two generators .