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I need a little help with the second half of the proof that $ (\mathbb{Z} \times \mathbb{Z},+) / \langle(2,3)\rangle $ is cyclic.

I know isomorphism preserves cyclic structure.I looked up the answer on my textbook from where I got this exercise and their proof is like this:

"The function $f : \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} /\langle(2,3)\rangle$ with $f(x) = \widehat{(x,x)}$ is an isomorphism of groups because :

  • $(x,x)$ is not in $\langle(2,3)\rangle$ for every $x \neq 0$.
  • $(a,b) =(b-a)(2,3)+(3a-2b)(1,1)$ for a,b in $\mathbb{Z}$ .

I think in the first part they try to show that the function is injective but I don't understand the second part,can someone explain it to me? Thanks alot !

Thomas Andrews
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Eduard Valentin
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    The second part is about showing surjectivity : any $(a,b)$ is equivalent (mod $\langle (2,3) \rangle$ ) to $(3a-2b) \cdot (1,1)$, which is in the image of $f$ (it's $f(3a-2b)$). Therefore every residue class mod $\langle (2,3) \rangle$ is in the image of $f$ : $f$ is surjective – Maxime Ramzi Jun 04 '17 at 14:44
  • Damn,thank you , now I understand, wasn't hard at all :\ . – Eduard Valentin Jun 04 '17 at 15:05
  • For $k=gcd(m,n)=1$ we have $(\mathbb{Z}\times \mathbb{Z})/\langle (m,n)\rangle\cong \mathbb{Z}$. – Dietrich Burde Jun 04 '17 at 16:01

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