Productivity of the Zariski topology on groups

This paper investigates the productivity of the Zariski topology 3(G) of a group G. If G {G(i) vertical bar i is an element of I} is a family of groups, and G = Pi(i is an element of I) G(i), is their direct product, we prove that 3(G) subset of Pi(i subset of I) 3(Gi). This inclusion can be proper in general, and we describe the doubletons G = {G(1),G(2)} of abelian groups, for which the converse inclusion holds as well, i.e., 3(G) = 3(G1) x 3(G2). If e(2) is an element of G(2) is the identity element of a group G(2) we also describe the class of groups G(2) such that G(1) x {e(2)} is an elementary algebraic subset of G(1) X G(2) for every group G(1). We show among others, that Delta is stable under taking finite products and arbitrary powers and we describe the direct products that belong to Delta. In particular, Delta contains arbitrary direct products of free non-abelian groups.