# Construction of a Finite Abelian Group

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!

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