Nullstellensatz for relative existentially closed groups

Abstract

We prove that in every variety of $G$-groups, every $G$-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize {\bf Theorem G} of \cite{BMR1}. As a result we see that every pair of $G$-existentially closed elements in an arbitrary variety of $G$-groups generate the same quasi-variety and if both of them are $q_{\omega}$-compact, they are geometrically equivalent.