On endomorphisms of free groups that preserve primitivity

Abstract.

It is proven that if $\phi$ is an endomorphism of a free group $F_n = \langle x_1, \dots , x_n \rangle$ of rank n such that $\phi (u)$ is primitive whenever so is $u \in$ F _n and $\phi$ (F _n ) contains a primitive pair (i.e., a pair $\alpha (x_1), \alpha (x_2)$ with $\alpha\in$ Aut F _n ), then $\phi$ is an automorphism. Also, every endomorphism of F ₂ that preserves primitivity is an automorphism.