I am interested in understanding all the subgroups of $\mathbb{Z}^{3}:=\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$.
$\mathbb{Z}^{3}$ a free abelian group of rank three, so all subgroups are free abelian of rank at most three. They are thus all isomorphic to either the trivial group, $\mathbb{Z}$, $\mathbb{Z}^{2}$ or $\mathbb{Z}^{3}$ itself. This is easy enough, but I want to understand the explicit subgroups, not just up to isomorphism.
Clearly all subgroups are generated by $m\leq 3$ linearly independent vectors in $\mathbb{Z}^{3}$, but I can't seem to come up with a nice way to describe these sets of vectors. Any pointers would be much appreciated.
There is nothing special about dimension 3.
Let $G \le {\mathbb Z}^n$, and suppose that $G$ is spanned by the linearly independent vectors $e_1,\ldots, e_m$ with $k \le n$.
Form an $m \times n$ matrix in which these vectors are the rows, then put that matrix into (row) Hermite Normal Form. Then the rows $f_1,\ldots,f_m$ of that new matrix provide a canonical set of generators for $G$.