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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 $\mathbb{Z}$.

Shaun
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blackkay1
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    Have Smith normal forms been covered in your course already? – Jyrki Lahtonen Dec 08 '21 at 19:14
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    Look at the matrix $$\pmatrix{3&1\cr8&2\cr}.$$ The gcd of the entries is $d_1=1$ and the gcd of $2\times2$ minors is $d_1d_2=2$. Therefore using Smith normal forms tells us that the quotient $\simeq\Bbb{Z}{d_1}\oplus\Bbb{Z}{d_2}\simeq \Bbb{Z}_1\oplus\Bbb{Z}_2\simeq\Bbb{Z}_2$. – Jyrki Lahtonen Dec 08 '21 at 19:19
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    Stacked bases theorem is a good buzzword for searching. – Jyrki Lahtonen Dec 08 '21 at 19:20
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    Here the simplest way is undoubtedly to show that ${(3,1),(4,1)}$ is a basis of $\Bbb{Z}\times\Bbb{Z}$. That makes the quotient somewhat clear (and we have found a pair of stacked bases). – Jyrki Lahtonen Dec 08 '21 at 19:23
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    Also note that the method used in the linked question is correct only because of the very simple choice of generators. – freakish Dec 08 '21 at 19:25
  • @JyrkiLahtonen Yes, in Algebra II two semesters ago. – Parcly Taxel Dec 08 '21 at 19:25
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    The method works, @freakish; see my answer below. – Shaun Dec 09 '21 at 13:44
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    I would even accept your answer Shaun, since it's 'more in line' with the other quention linked to this one, but I have already accepted the answer of Parcly Taxel (which is also great). – blackkay1 Dec 09 '21 at 15:09
  • You can still accept it, @blackkay1; it just takes it away from ParclyTaxel – Shaun Dec 09 '21 at 15:12

2 Answers2

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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: $$\begin{bmatrix}3&1\\8&2\end{bmatrix}\to\begin{bmatrix}3&1\\2&0\end{bmatrix}\to\begin{bmatrix}1&1\\2&0\end{bmatrix}\to\begin{bmatrix}0&1\\2&0\end{bmatrix}\to\begin{bmatrix}1&0\\0&2\end{bmatrix}$$

Parcly Taxel
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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 $x$.

Shaun
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