Limit groups and automorphisms of κ-existentially closed groups

Date

2025-03-15

Author

Kaya, Burak
Kuzucuoğlu, Mahmut
Longobardi, Patrizia
Maj, Mercede

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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.

Subject Keywords

Algebraically closed groups, Automorphism groups, Existentially closed groups

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https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85212544589&origin=inward
https://hdl.handle.net/11511/113212

Collections

Department of Mathematics, Article