EXISTENTIALLY AND κ-EXISTENTIALLY CLOSED GROUPS

Date

2023-03-01

Author

Kaya, Burak
Kuzucuoğlu, Mahmut

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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.

Subject Keywords

algebraically closed groups, automorphism groups, Existentially closed groups

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https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85133926465&origin=inward
https://hdl.handle.net/11511/102951

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Department of Mathematics, Article