$m'$-group being cyclic?

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?

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.

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