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Category:Tin pan production Category:History of chemistry Category:Anti-fracking movement Category:Mottos Category:Tin Category:PackagingQ: If $H$ is a subgroup of $G$ with order $lcm(n,m)$ then $H$ is cyclic. Let $n,m$ be two natural numbers. Show that if $H$ is a subgroup of $G$ with order $lcm(n,m)$ then $H$ is cyclic. A: Suppose $lcm(n,m)$ is not $n$ or $m$. Then the subgroup is isomorphic to the free abelian group generated by $n$ and $m$ and has rank $2$. The subgroup is of order $lcm(n,m)$ by hypothesis and as a free abelian group is generated by $n$ and $m$. The rank of a free abelian group is the number of basis elements, which is the same as the rank of a vector space.
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