# If $H$ is a subgroup of $\mathbb Q$ then $\mathbb Q/H$ is infinite

I'm trying to work out this question:

Prove that if $H$ is a proper subgroup of $\mathbb{Q}$ then $\mathbb{Q}/H$ is infinite, but each of its elements have finite order.

I thought, for the first part, that I could assume for contradiction that $\mathbb{Q}/H$ is finite of order $n$, then for all $\dfrac{a}{b}\in\mathbb{Q}$, $\dfrac{a^n}{b^n}$ is in $H$. And somehow prove this is a contradiction. Any help?

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2022-07-25 20:47:17
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