This paper investigates the productivity of the Zariski topology ${ℨ}_G$ of a group $G$. If $𝒢=\{G_i\mid i\in I\}$ is a family of groups, and $G={\prod }_{i\in I}{G}_i$ is their direct product, we prove that ${ℨ}_G\subseteq {\prod }_{i\in I}{ℨ}_{G_i}$. 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., ${ℨ}_G={ℨ}_{G_1}\times {ℨ}_{G_2}$. 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 $G_1\times \left\{e_2\right\}$ is an elementary algebraic subset of $G_1\times G_2$ for every group $G_1$. We show among others, that $\Delta$ is stable...