A question of B. Plotkin about the semigroup of endomorphisms of a free group

Abstract

Let F F be a free group of finite rank n ≥ 2 n \geq 2 , let E n d ( F ) End(F) be the semigroup of endomorphisms of F F , and let A u t ( F ) Aut(F) be the group of automorphisms of F F . Theorem. If T : E n d ( F ) → E n d ( F ) T : End(F) \to End(F) is an automorphism of E n d ( F ) End(F) , then there is an α ∈ A u t ( F ) \alpha \in Aut(F) such that T ( β ) = α ∘ β ∘ α − 1 T(\beta ) = \alpha \circ \beta \circ \alpha ^{-1} for all β ∈ E n d ( F ) \beta \in End(F) .