Automorphisms of κ -existentially closed groups

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Monatshefte fur Mathematik, vol.198, no.4, pp.791-804, 2022 (Journal Indexed in SCI Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 198 Issue: 4
  • Publication Date: 2022
  • Doi Number: 10.1007/s00605-022-01730-0
  • Title of Journal : Monatshefte fur Mathematik
  • Page Numbers: pp.791-804
  • Keywords: Algebraically closed groups, Existentially closed groups


© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.We investigate the automorphisms of some κ-existentially closed groups. In particular, we prove that Aut(G) is the union of subgroups of level preserving automorphisms and | Aut(G) | = 2 κ whenever κ is inaccessible and G is the unique κ-existentially closed group of cardinality κ. Indeed, the latter result is a byproduct of an argument showing that, for any uncountable κ and any group G that is the limit of regular representation of length κ with countable base, we have | Aut(G) | = ℶκ+1, where ℶ is the beth function. Such groups are also κ-existentially closed if κ is regular. Both results are obtained by an analysis and classification of level preserving automorphisms of such groups.