Let $H$ and $K$ two subgroups of the additive group of rational numbers $( \mathbb{Q},+,-,0)$. Show that if there are positive integers $m$ and $n$ such that $mH \subset K$ and $nK \subset H$ then $H\cong K$.
2026-03-27 07:18:45.1774595925
isomorphic subgroups of the additive group of rational numbers
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Hint: A subgroup of $\Bbb Q$ is uniquely determined by the prime powers it contains.