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Consider the group $\mathbb{Q}$ under addition of rational numbers. If $H$ is a subgroup of $\mathbb{Q}$ with finite index, then $H = \mathbb{Q}$.

I just saw this on our exam earlier and was stumped on how to show this. Any ideas?

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    You might want to look at the quotient group $\mathbb Q / H$ and note that it's finite.2012-08-14

3 Answers 3