THE TWISTED CONJUGACY PROBLEM FOR ENDOMORPHISMS OF METABELIAN GROUPS

Let M be a finitely generated metabelian group explicitly presented in a variety A(2) of all metabelian groups. An algorithm is constructed which, for every endomorphism phi is an element of End(M) identical modulo an Abelian normal subgroup N containing the derived subgroup M' and for any pair of elements u, v is an element of M, decides if an equation of the form (x phi)u = vx has a solution in M. Thus, it is shown that the title problem under the assumptions made is algorithmically decidable. Moreover, the twisted conjugacy problem in any polycyclic metabelian group M is decidable for an arbitrary endomorphism phi is an element of End(M).