University of Isfahan International Journal of Group Theory 2251-7650 12 1 2023 03 01 Existentially and $\kappa$-existentially closed groups 45 54 26352 10.22108/ijgt.2022.131513.1758 EN Burak Kaya Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey Mahmut Kuzucuoğlu Department of Mathematics, Middle East Technical University, 06800,Ankara, Turkey Journal Article 2021 11 17 A group $G$ is existentially closed (algebraically closed) if every finite system of equations and in-equations that has coefficients in $G$ and has a solution in an overgroup $H\geq G$ has a solution in $G$. Existentially closed groups were introduced by W. R. Scott in 1951. B. H. Neumann posed the following question in 1973: Does there exist explicit examples of existentially closed groups? Generalized version of this question is as follows: Let $\kappa$ be an infinite cardinal. Does there exist explicit examples of $\kappa$-existentially closed groups? Recently an affirmative answer was given to Neumann's question and the generalized version of it, by Kaya-Kegel-Kuzucuo\u{g}lu. We give a survey of these results. We also prove that, there are maximal subgroups of $\kappa$-existentially existentially closed groups and provide some information about subgroups containing the centralizer of subgroups generated by fewer than $\kappa$-elements. This generalizes a result of Hickin-Macintyre. https://ijgt.ui.ac.ir/article_26352_bb5dd75b306e31d26a0a4627b820259e.pdf