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## Isomorphic to $(\mathbb Z \times\mathbb Z )\,\big/\langle(1,2)\rangle$

Groups, Rings, Domains, Modules, etc, Galois theory
Grigorios Kostakos
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### Isomorphic to $(\mathbb Z \times\mathbb Z )\,\big/\langle(1,2)\rangle$

Which group is isomorphic to the quotient group $({\mathbb{Z}}\times{\mathbb{Z}})\big/\langle{(1,2)}\rangle$ , where $\langle{(1,2)}\rangle$ is the normal subgroup of ${\mathbb{Z}}\times{\mathbb{Z}}$ generated by $(1,2)$ ? Justify your answer.
Grigorios Kostakos

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Grigorios Kostakos
Founder
Articles: 0
Posts: 460
Joined: Mon Nov 09, 2015 1:36 am
Location: Ioannina, Greece

### Re: Isomorphic to $(\mathbb Z \times\mathbb Z )\,\big/\langle(1,2)\rangle$

We give a solution:

The function $f:{\mathbb{Z}}\times{\mathbb{Z}}\longrightarrow\mathbb{Z}$ defined as $f((m,n))=2m-n\,,$ it is a well defined epimorphism (surjective homomorphism) with \begin{align*}
\ker{f}&=\bigl\{{(m,n)\in\mathbb{Z}\times{\mathbb{Z}}:f((m,n))=0}\bigr\}\\
&=\bigl\{{(m,n)\in\mathbb{Z}\times{\mathbb{Z}}:n=2m}\bigr\}\\
&=\bigl\{{(m,2m):m\in\mathbb{Z}}\bigr\}\\
&=\bigl\langle{(1,2)}\bigr\rangle\,.
\end{align*}
So, by the 1st Isomorphism theorem we have that $({\mathbb{Z}}\times{\mathbb{Z}})\big/\bigl\langle{(1,2)}\bigr\rangle\cong\mathbb{Z}\,.$
Grigorios Kostakos