University of Isfahan International Journal of Group Theory 2251-7650 11 2 2022 06 01 Nullstellensatz for relative existentially closed groups 125 130 25702 10.22108/ijgt.2021.125453.1652 EN Mohammad Shahryari Department of Mathematics, College of Science, Sultan Qaboos University, Muscat, Oman Journal Article 2020 10 16 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  <strong>Theorem G</strong> of [<em>J‎. ‎Algebra,</em>  <strong>219</strong> (1999) ‎16--79]‎. ‎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. ‎Algebraic geometry over groups‎ ‎Nullstellensatz‎ ‎Existentially closed groups‎ ‎Varieties of groups‎ ‎Quasi-varieties https://ijgt.ui.ac.ir/article_25702_c4dfb33866e9635cd387f013c94619a8.pdf