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 ≥ 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 κ be an infinite cardinal. Does there exist explicit examples of κ-existentially closed groups? Recently an affirmative answer was given to Neumann’s question and the generalized version of it, by Kaya-Kegel-Kuzucuoğlu. We give a survey of these results. We also prove that, there are maximal subgroups of κ-existentially existentially closed groups and provide some information about subgroups containing the centralizer of subgroups generated by fewer than κ-elements. This generalizes a result of Hickin-Macintyre.
International Journal of Group Theory
Citation Formats
B. Kaya and M. Kuzucuoğlu, “EXISTENTIALLY AND κ-EXISTENTIALLY CLOSED GROUPS,” International Journal of Group Theory, vol. 12, no. 1, pp. 45–54, 2023, Accessed: 00, 2023. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85133926465&origin=inward.