Is Thompson's group F residually finite?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
The answer is "no"  the quickest way to see this appeals to the following nontrivial fact: the commutator subgroup of $F$, denoted by $F'$ as usual, is infinite and simple. Armed with this, we argue as follows. Let $N$ be a normal subgroup in $F$ of finiteindex; then $N\cap F'$ is going to be normal in $F'$ and of finite index in $F'$. Hence $N\cap F'=F'$, that is, $N$ contains $F'$. So the intersection of all finiteindex normal subgroups of $F$ must contain $F'$. But if $F$ were residually finite then this intersection would only contain the identity element, and the result follows. If you don't mind me asking: is this a question out of curiosity, or one that you've run into during your studies or research? 

