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

We give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. Using this characterization, we 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 of two such groups G,H is equivalent to each of the following properties: 1)Gx…xG (k times G) and Hx…xG (k times H) are elementarily equivalent for a strictly positive integer k; 2)AxG and AxH are elementarily equivalent for two elementarily equivalent polycyclic-by-finite groups A,B. It is not presently known if 1) implies elementary equivalence for any groups G,H.