On Primitive and Inner Endomorphisms of Groups

Abstract

We consider the semigroup $${\cal P}(G)$$ of primitive endomorphisms with respect to a free group G for which the group Aut(G) coincides with the group of tame automorphisms TAut(G). We find the necessary and sufficient conditions for a given automorphism to belong to $${\cal P}(G)$$ and also the necessary and sufficient conditions for the coincidence of $${\cal P}(G)$$ and Aut(G). Considering a free metabelian group G, we prove the quasi-identity $$\varphi \psi \in {\cal P}(G) \Rightarrow \psi \in {\cal P}(G).$$. Some properties of inner endomorphisms of metabelian groups are established, and a few corollaries is obtained.