I'm trying to work through the following problem.
Let $M=(m_{ij})$ be a $3\times 3$ matrix with integer entries. Assume that det$(M)\neq 0$. Consider the group homomorphism $f:\mathbb{Z}^{3}\to\mathbb{Z}^{3}$ by $f(a,b,c)=M(a,b,c)^{T}$. Prove that the quotient group $\mathbb{Z}^{3}/f(\mathbb{Z}^{3})$ is a finite abelian group of order $|\mathrm{det}(M)|$.
I tried reading through my textbook (Lang's Algebra), but I didn't see anything that I thought would be particularly helpful. I haven't tried anything yet because I don't even know what to try as a first step. Any advice is greatly appreciated!