Existentially and $\kappa$-existentially closed groups

Abstract

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.

Keywords

• existentially closed groups
• algebraically closed groups
• automorphism groups