EXISTENTIALLY AND EXISTENTIALLY CLOSED GROUPS

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.