On the existence of kappa-existentially closed groups

Kaya B., Kegel O. H., Kuzucuoglu M.

ARCHIV DER MATHEMATIK, vol.111, pp.225-229, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 111
  • Publication Date: 2018
  • Doi Number: 10.1007/s00013-018-1213-x
  • Journal Name: ARCHIV DER MATHEMATIK
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.225-229
  • Keywords: Existentially closed groups, Infinite symmetric groups, Homogeneous groups
  • Middle East Technical University Affiliated: Yes

Abstract

We prove that a κ-existentially closed group of cardinality λ exists whenever κ ≤ λ are uncountable cardinals with λ^{<κ} = λ. In particular, we show that there exists a κ-existentially closed group of cardinality κ for regular κ with 2^{<κ} = κ. Moreover, we prove that there exists no κ-existentially closed group of cardinality κ for singular κ. Assuming the generalized continuum hypothesis, we completely determine the cardinals κ ≤ λ for which a κ-existentially closed group of cardinality λ exists