Limit Groups and Automorphisms of $\kappa$-Existentially Closed Groups
Burak Kaya, Mahmut Kuzucuo\u{g}lu, Patrizia Longobardi, Mercede Maj

TL;DR
This paper investigates the automorphism groups of $oldsymbol{ ext{kappa}}$-existentially closed groups, establishing their cardinality as $2^{ ext{kappa}}$ for groups of size $ ext{kappa}$ and confirming the existence of universal groups for any uncountable $ ext{kappa}$.
Contribution
It proves that automorphism groups of $oldsymbol{ ext{kappa}}$-existentially closed groups of size $ ext{kappa}$ have size $2^{ ext{kappa}}$ and affirms the existence of universal groups for all uncountable cardinals.
Findings
Automorphism group size is $2^{ ext{kappa}}$ for $ ext{kappa}$-existentially closed groups of size $ ext{kappa}$.
Confirmed the existence of universal groups of size $ ext{kappa}$ for any uncountable $ ext{kappa}$.
Extended previous results on automorphism groups of such groups.
Abstract
The structure of automorphism groups of -existentially closed groups are studied by Kaya-Kuzucuo\u{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and whenever is an inaccessible cardinal and is the unique -existentially closed group of cardinality . The cardinality of the automorphism group of a -existentially closed group of cardinality is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case namely: If is a -existentially closed group of cardinality , then . We also answer Kegel's question on universal groups, namely: For any uncountable cardinal , there exist universal groups of cardinality .
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Topology and Set Theory · Geometric and Algebraic Topology
