Existentially and $\kappa$-existentially closed groups

Kaya, Burak; Kuzucuoğlu, Mahmut

doi:10.22108/ijgt.2022.131513.1758

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	Existentially and $\kappa$-existentially closed groups
International Journal of Group Theory
مقاله 6، دوره 12، شماره 1، خرداد 2023، صفحه 45-54 اصل مقاله (428.06 K)
نوع مقاله: Ischia Group Theory 2020/2021
شناسه دیجیتال (DOI): 10.22108/ijgt.2022.131513.1758
نویسندگان
Burak Kaya¹؛ Mahmut Kuzucuoğlu^* ²
¹Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey
²Department of Mathematics, Middle East Technical University, 06800,Ankara, Turkey
چکیده
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.
کلیدواژه‌ها
Existentially closed groups؛ algebraically closed groups؛ automorphism groups

مراجع
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Existentially and $\kappa$-existentially closed groups