Recall that a group $G$ is Hopfian if every epimorphism $f :G\longrightarrow G$ is an automorphism (equivalently, $N=1$ is the only normal subgroup for which $G/N\cong G$).
Is a free group of finite rank Hopfian?
Thanks in advance.
You can prove directlty that the free group $F_n$ of finite rank $n$ is Hopfian by using the theory of Nielsen transformations. Any subset $S$ of $F_n$ can be transformed to a set of free generators of $\langle S \rangle$ by applying a succession of these transformations. One type of transformation is to delete an element of $S$ that is equal to the identity element. The others all transform one free generating set of $S$ to another.
Suppose that $F_n$ is freely generated by $a_1,\ldots,a_n$ and $\phi:F_n \to F_n$ is an epimorhism with $\phi(a_i) = b_i$. Now apply Nielsen transformations to $\{b_1,\ldots,b_n\}$, which by assumption generate $F_n$. It is easy to see that $F_n$ cannot be generated by fewer than $n$ elements, so none of the transformaions applied can delete an element. This means that $\{b_1,\ldots,b_n\}$ was already a free generating set of $F_n$, so $\phi$ has trivial kernel, and hence $F_n$ is Hopfian.