Nullstellensatz for relative existentially closed group

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  Theorem G of [J‎. ‎Algebra,  ۲۱۹ (۱۹۹۹) ‎۱۶--۷۹]‎. ‎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.

Keywords:

‎Algebraic geometry over groups‎ , ‎Nullstellensatz‎ , ‎Existentially closed groups‎ , ‎Varieties of groups‎ , ‎Quasi-varieties