Automorphic and endomorphic reducibility and primitive endomorphisms of free metabelian groups

Abstract

Let S be the free metabelian group of rank 2. In this paper we prove the following results:(i) Given a pair of elements g, h of S, there exists an algorithm to decide whether or not g is an automorphic image of h; (ii) If g, h are in the commutator subgroup S′ of S such that each is an endomorphic image of the other then g , h are automorphic; (iii) If an endomorphism of S maps primitive elements of S to primitive elements of S then it defines an automorphism of S. We also include an example to show that, in (ii) above,the requirement that g,h are in S ? can not be relaxed.