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I was looking at a paper, and I saw this claim,

It is obvious that if $H$ has finite index in $F_m$ then $H$ has non-trivial intersection with each of the non-trivial subgroups of $F_m$.

Why is this immediate?

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Because $H$ intersects each subgroup in a finite index subgroup (bounded above by the index of $H$ in $F_m$) and because $F_m$ has no finite subgroups. –  Alex B. Dec 19 '10 at 6:30
    
Thanks, missed that. –  Bazooka Dec 19 '10 at 6:40
    
This is true, and easy to prove, for any torsion-free group. –  Angelo Dec 19 '10 at 6:41
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