TITLE: Does every group have a finite index subgroup?
QUESTION [5 upvotes]: Does every group have a finite index subgroup (besides the whole group)? I suspect the answer is no, but I haven't found an example.
What about finitely presented or generated groups? Or groups with torsion?
I tried something like taking a presentation and only selecting some of the generators, but I don't this works even if $G$ has torsion, since the other generators could interact in a complicated way. Perhaps if we could find a minimal presentation of a group with torsion containing (as a generator) a torsion element, then this would work, but I'm not sure why such a minimal presentation should exist.

REPLY [5 votes]: The condition that a group has a nontrivial finite index subgroup is equivalent to the condition that it has a nontrivial finite quotient, and also equivalent to the condition that it acts nontrivially on some finite set (exercise). 
As stated in the comments, $\mathbb{Q}$ is an easy example of a group with no nontrivial finite quotients (exercise), the point being that quotients of divisible groups are divisible but no nontrivial element of a finite group is divisible. If you want an example which is torsion then take $\mathbb{Q}/\mathbb{Z}$, which is still divisible.
Even assuming finitely presented does not help you: the Higman group is a finitely presented group with no nontrivial finite quotients.