Existentially and $\kappa$-existentially closed groups

Document Type : Ischia Group Theory 2020/2021

Authors

1 Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

2 Department of Mathematics, Middle East Technical University, 06800,Ankara, Turkey

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.

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References

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International Journal of Group Theory
Volume 12, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory (2020/2021) - Part 1
March 2023
Pages 45-54

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History

  • Receive Date: 17 November 2021
  • Revise Date: 08 February 2022
  • Accept Date: 16 February 2022
  • Published Online: 01 March 2023

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