Limit groups and automorphisms of κ-existentially closed groups

KAYA B., KUZUCUOĞLU M., Longobardi P., Maj M.

Journal of Algebra, vol.666, pp.840-849, 2025 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 666
  • Publication Date: 2025
  • Doi Number: 10.1016/j.jalgebra.2024.12.003
  • Journal Name: Journal of Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.840-849
  • Keywords: Algebraically closed groups, Automorphism groups, Existentially closed groups
  • Middle East Technical University Affiliated: Yes

Abstract

The structure of automorphism groups of κ-existentially closed groups has been studied by Kaya-Kuzucuoğlu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and |Aut(G)|=2κ whenever κ is an inaccessible cardinal and G is the unique κ-existentially closed group of cardinality κ. The cardinality of the automorphism group of a κ-existentially closed group of cardinality λ>κ is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case κ=λ namely: If G is a κ-existentially closed group of cardinality κ, then |Aut(G)|=2κ. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal κ, there exist universal groups of cardinality κ.