Classify $\mathbb{Z} \times \mathbb{Z}/ \langle (3,1), (8,2) \rangle $ via fundamental theorem of finitely generated abelian groups

I am trying to classify $\mathbb{Z} \times \mathbb{Z}/ \langle (3,1), (8,2) \rangle $ via fundamental theorem of finitely generated abelian groups. As I understand by different questions posted in here that's what I'm doing.

I started to research that topic by posting that question and then got interested in different possible options.

According to the scheme provided by Shaun I would think that I need to write $x = (1,0)$ and $y = (0,1)$. By that I would obtain $(3,1) = 3x + y$ and $(8,2) = 8x + 2y$. But what to do next? As I understand, it is not isomorphic to any $\mathbb{Z}$.

Solution 1:

The algorithmic way of solving problems like this is to compute the Smith normal form of the matrix formed by the ideal generators, which reveals the invariant factors. In this case the invariant factors are $1,2$, which means the group is isomorphic to $\mathbb Z/(1)\oplus\mathbb Z/(2)=\mathbb Z_2$.

The following invertible elementary row/column operations convert the original matrix to Smith normal form: $$\begin{bmatrix}3&1\\8&2\end{bmatrix}\to\begin{bmatrix}3&1\\2&0\end{bmatrix}\to\begin{bmatrix}1&1\\2&0\end{bmatrix}\to\begin{bmatrix}0&1\\2&0\end{bmatrix}\to\begin{bmatrix}1&0\\0&2\end{bmatrix}$$

Solution 2:

Let $f$ be as defined in the question. We get

$$\begin{align} \Bbb Z\times \Bbb Z/\ker f &\cong \langle x,y\mid 3x + y, 8x + 2y, x+y=y+x\rangle\\ &\cong\langle x,y\mid y=-3x, 8x+2y, x+y=y+x\rangle\\ &\cong\langle x\mid 8x-6x, x-3x=-3x+x\rangle\tag{1}\\ &\cong\langle x\mid 2x, -2x=-2x\rangle\\ &\cong\langle x\mid 2x\rangle\\ &\cong \Bbb Z_2, \end{align}$$

where $(1)$ holds by the Tietze transformation of eliminating $y$ as $y$ can be written in terms of $x$.