Existentially and $\kappa$-existentially closed groups
| Autor: | Burak Kaya, Mahmut Kuzucuoğlu |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | International Journal of Group Theory, Vol 12, Iss 1, Pp 45-54 (2023) |
| Druh dokumentu: | article |
| ISSN: | 2251-7650 2251-7669 |
| DOI: | 10.22108/ijgt.2022.131513.1758 |
| Popis: | 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. |
| Databáze: | Directory of Open Access Journals |
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