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 \(u \in \)Fn and \(\phi\) (Fn) contains a primitive pair (i.e., a pair \(\alpha (x_1), \alpha (x_2)\) with \(\alpha\in \) Aut Fn), then \(\phi \) is an automorphism. Also, every endomorphism of F2 that preserves primitivity is an automorphism.