EXISTENTIALLY AND κ-EXISTENTIALLY CLOSED GROUPS


KAYA B., KUZUCUOĞLU M.

International Journal of Group Theory, vol.12, no.1, pp.45-54, 2023 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 12 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.22108/ijgt.2022.131513.1758
  • Journal Name: International Journal of Group Theory
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, Academic Search Premier, zbMATH, Directory of Open Access Journals
  • Page Numbers: pp.45-54
  • Keywords: algebraically closed groups, automorphism groups, Existentially closed groups
  • Middle East Technical University Affiliated: Yes

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 ≥ 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.