On primitive and inner endomorphisms of groups (Q2191366)

Let \(G_r\) be a relatively free group of rank \(r\) in some variety of groups \(V\). An element \(g\in G_r\) is called primitive if it contains in some basis of \(G_r\). An endomorphism \(\phi : G_r \rightarrow G_r\) is said to preserve primitive elements or, briefly, primitive if the image \(\phi (g)\) of each primitive element \(g\) is a primitive element. Denote by P(\(G_r\)) the set (monoid) of all primitive endomorphisms of \(G_r\). Let the automorphism group Aut(\(G_r\)) coincides with the group of tame automorphisms TAut(\(G_r\)). Then the author finds the necessary and sufficient conditions for a given automorphism to be a primitive endomorphism. He also derives the necessary and sufficient conditions for the coincidence of Aut(\(G_r\)) with P(\(G_r\)). Considering the free metabelian group \(M_r\), the author proves the quasi-identity \(\phi \cdot \psi \in P(M_r) \rightarrow \psi \in P(M_r).\) Some properties of inner endomorphisms of groups \(M_r\) are also established.