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!

  • $\begingroup$ Have you tried computing examples? What happens in the case where $M$ is the identity? $\endgroup$ – Santana Afton Oct 15 '18 at 16:44
  • 1
    $\begingroup$ See math.stackexchange.com/a/2768134/589 $\endgroup$ – lhf Oct 15 '18 at 16:46
  • $\begingroup$ @SantanaAfton So in that case, we mod out by the vector $(a,b,c)^{T}$, which should just give us the identity, right? $\endgroup$ – Sir_Math_Cat Oct 15 '18 at 16:48

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