A question about a free abelian finitely generated group.

Question

I am having a hard time solving this and it is really confusing. I don't have enough schema, which makes it problematic. Let $A$ be a finitely generated free abelian group and $B$ is a subgroup of $A.$ Assume there is a basis $\{f_1,f_2\}$ of $A$ such that $\{f_1,mnf_2\}$ is a basis of $B$, whereas $(m,n)=1.$ I have to prove there is a basis $\{g_1,g_2\}$ of $A$ such that $\{ng_1,mg_2\}$ is a basis of $B.$ How can I show that? Thank you for your help.

Answer 1

You have to use the fact that $(m, n) = 1$. This means there exists integers $a, b \in \mathbb Z$ such that $an + bm = 1$. One way to think of bases is they are the columns of matrices in $\mathrm{GL}_2(\mathbb Z)$. So you look for a way to write down a $2 \times 2$ matrix with determinant $\pm1$ using the integers $a, b, n, m$. My first guess was $$\begin{bmatrix} a & b \\ m & -n \end{bmatrix}$$ and this corresponds to the basis $\{af_1 + mf_2, bf_1 - nf_2\}$. I'll leave you to check that letting these be $g_1$ and $g_2$ works.

I wish I could say something more insightful about how to come up with this answer. Having a limited amount of information means there are a very limited number of things you can actually try, so you just follow your nose and start trying them.