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I am just starting a new topic in group theory and I want to understand what is a group extension. I have the following exercise (probably simple): Check that $Z_{mn}$ is an extension of $Z_{m}$ by $Z_{n}$. I want to know what is the method of solving these kind of problems.

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This means you need to show that there is an exact sequence $0 \to \Bbb{Z}_n \to \Bbb{Z}_{mn} \to \Bbb{Z}_m \to 0$. Take for instance the case $n=3$ and $m=2$. Then the first map $\Bbb{Z}_3 \to \Bbb{Z}_6$ is the inclusion of the subgroup $\left<2\right> = \{0,2,4\} \subset \Bbb{Z}_6$, which is isomorphic to $\Bbb{Z}_3$. The second map $\Bbb{Z}_6 \to \Bbb{Z}_2$ is then just the quotient by this subgroup.

In general, to show that $G$ is an extension of $Q$ by $N$, you need to find a normal subgroup $N \unlhd G$ so that $G/N \cong Q$.

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