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I'm trying to prove the following: if $F$ is a free group and $H$ is a subgroup of $F$ such that the index $[F:H]$ is finite, then $H \cap K \neq 1$ for every nontrivial subgroup $K$ of $F$.

I don't know where to begin so I would much appreciate a hint.

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Hint: If $K$ is a subgroup of $F$ such that $H \cap K = 1$, then any two distinct elements of $K$ lie in distinct cosets of $H$. (You can use left or right cosets here, doesn't matter.)

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+1 This is the kind of answer I like to see! Too many people provide more or less a full answer even when only a hint is asked for. – Tara B Feb 17 '13 at 12:57

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