Summary: This paper investigates the productivity of the Zariski topology \(\mathfrak Z_G\) of a group \(G\). If \(\mathcal G=\{G_i\mid i\in I\}\) is a family of groups and \(G=\prod _{i\in I}G_i\) is their direct product, we prove that \(\mathfrak Z_G\subseteq\prod _{i\in I}\mathfrak Z_{G_i}\). This inclusion can be proper in general and we describe the doubletons \(\mathcal G=\{G_1,G_2\}\) of Abelian groups, for which the converse inclusion holds as well, i.e., \(\mathfrak Z_G=\mathfrak Z_{G_1}\times\mathfrak Z_{G_2}\). If \(e_2\in G_2\) is the identity element of a group \(G_2\), we also describe the class \(\Delta\) of groups \(G_2\) such that \(G_1\times\{e_2\}\) is an elementary algebraic subset of \(G_1\times G_2\) 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.