Determine the isomorphism class of M/T(M)

Question

Let $M=\Bbb{Z}\oplus\Bbb{Z}\oplus\Bbb{Z}$ and $T: M\rightarrow M$ given by $T(x,y,z)=(4x+2z,2y,2x+10z)$. Show the cokernel $M/T(M)$ is an abelian group of order $72$, and determine its isomorphism class.

I tried to solve it by looking at when the linear system has solution over Z, but it didn't get me anywhere.
I feel like this should not be an awefully hard question, but I don't know how to start. Can you give me some help? Thank you very much.

Answer 1

$T(M)$ has generators $(4,0,2)$, $(0,2,0)$, $(2,0,10)$, so we have the relations matrix

$$\left( \begin{array}{ccc} 4 & 0 & 2 \\ 0 & 2 & 0 \\ 2& 0 & 10 \\ \end{array} \right)$$

Computing the Smith normal form for this matrix gives

$$\left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0& 0 & 18 \\ \end{array} \right)$$

and hence $M/T(M) \cong \mathbb{Z}/2 \oplus \mathbb{Z}/2 \oplus \mathbb{Z}/18$ which has order $72$.

Answer 2

If you try to solve $Ax=b$ where

$A= \left( \begin{array}{ccc} 4 & 0 & 2 \\ 0 & 2 & 0 \\ 2& 0 & 10 \\ \end{array} \right)$ and $b=\left(\begin{array}{c}b_{1}\\b_{2}\\b_{3}\end{array}\right)$

you need check which condition $b_{1}, b_{2}$ and $b_{3}$ must have in order to have a solution in $\mathbb{Z}^{3}$.

In this case $T(M)=36\mathbb{Z}\oplus 2\mathbb{Z}\oplus9\mathbb{Z}$

So

$\begin{eqnarray}\frac{M}{T(M)}&=&\frac{\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}}{36\mathbb{Z}\oplus 2\mathbb{Z}\oplus9\mathbb{Z}}\\ &\cong&\frac{\mathbb{Z}}{36\mathbb{Z}}\oplus\frac{\mathbb{Z}}{2\mathbb{Z}}\oplus\frac{\mathbb{Z}}{9\mathbb{Z}}\\ &\cong& \mathbb{Z}_{36}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{9} \end{eqnarray}$

I hope it helps you.