I'm never sure about free groups whether a question is easy or not. It feels to me like this is impossible, but I couldn't come up with any argument.


If $F\_n$ is a free group on $n$ generators, could there exist a non-trivial $N\triangleleft F\_n$ such that $F\_n/N \cong F\_n$?



Can a finitely generated free group be isomorphic to a non-trivial quotient of itself?
No, the free groups are [Hopfian](http://en.wikipedia.org/wiki/Hopfian_group) because they are [residually finite](http://en.wikipedia.org/wiki/Residually_finite_group). 


