So you have a free group $F_n$, freely generated by $\alpha_1 \cdots \alpha_n$. Pick any $n$ elements $g_1 \cdots g_n$ and define an endomorphism $\psi$ of $F_n$ by $\psi(\alpha_i) = g_i^{-1}\alpha_ig_i$ and extend as usual.

It looks very much to me that $\psi$ will always be injective, but I'm having a hard time proving this. Any ideas?

Cheers in advance.

Hair