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 $hgeq g$ 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-kuzucuou{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.