Direct products and elementary equivalence of polycyclic-by-finite groups

Generalizing previous results, we give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. We use this characterization to investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular, we prove that the elementary equivalence G≡H of two such groups G, H is equivalent to each of the following properties: (1) G×⋯×G(k times G)≡H×⋯×H(k times H) for an integer k≥1; (2) A×G≡B×H for two polycyclic-by-finite groups A, B such that A≡B. It is not presently known if (1) implies G≡H for any groups G, H.