Let $a, b, c, d, e, f, g$ denote integers, with $a$ and $c$ both non-zero, and consider the matrix
$$ M=\left(\begin{matrix} a&b&d&f \\ 0&c&e&g \end{matrix}\right) $$
Let $G$ be the quotient of the group $\Bbb Z^2$ by the column space of $M$. That is, $G$ is presented by two generators, say $x,y$, which commute, and which satisfy the four relations $ax=0$, $bx+cy=0$, $dx+ey=0$, $fx+gy=0$.
What is the order of $x$ in $G$, in terms of $a, b, c, d, e, f, g$?
This question is a challenge because I have an answer (in fact, I have two approaches that yield the same answer). The point of asking is that this problem has a huge teaching/learning potential, and to see whether there could be yet another approach.
Hint: the problem is elementary if you remove the last two columns. There is a twist if you add $d\choose e$, and yet quite another twist if you add the last column. Then, adding more columns would not make the problem more interesting. In fact, the problem given with a $2\times 4$ matrix as above is essentially as difficult as the same problem given with an arbitrary matrix.
