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Given a group $G=\mathbb{Z}_m\rtimes\mathbb{Z}_n$ with $m,n$ coprime. Should every subgroup of $G$ that has order coprime to $m$ be cyclic?

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Yes, because if $H$ is such a subgroup, then by Lagrange $H \cap \Bbb{Z}_{m} = \{ e \}$, so that $H$ is isomorphic to its image under the homomorphism $$ G \to \Bbb{Z}_{n} $$ that has $\Bbb{Z}_{m}$ as kernel. So $H$ is isomorphic to a subgroup of the cyclic group $\Bbb{Z}_{n}$, and as such $H$ is cyclic.

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  • $\begingroup$ Can we say that this $H$ is unique up to isomorphism? Thanks $\endgroup$ – mrs Mar 25 '13 at 8:33
  • $\begingroup$ @Babak.S, if it is cyclic, then it is automatically unique. $\endgroup$ – Easy Mar 25 '13 at 8:34
  • $\begingroup$ @BabakS., there are as many isomorphism types for $H$ as there are distinct subgroups of $\Bbb{Z}_{n}$. $\endgroup$ – Andreas Caranti Mar 25 '13 at 8:35
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    $\begingroup$ @BabakS., we can say that the isomorphism type of $H$ is determined by its order. $\endgroup$ – Andreas Caranti Mar 25 '13 at 8:36
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    $\begingroup$ @BabakS., you're welcome, and nice to see you :-) $\endgroup$ – Andreas Caranti Mar 25 '13 at 8:36

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