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On the existence of \(\kappa \)-existentially closed groups. (English) Zbl 1498.20003

Summary: We prove that a \(\kappa \)-existentially closed group of cardinality \(\lambda \) exists whenever \(\kappa \leq \lambda \) are uncountable cardinals with \(\lambda ^{<\kappa }=\lambda \). In particular, we show that there exists a \(\kappa \)-existentially closed group of cardinality \(\kappa \) for regular \(\kappa \) with \(2^{<\kappa }=\kappa \). Moreover, we prove that there exists no \(\kappa \)-existentially closed group of cardinality \(\kappa \) for singular \(\kappa \). Assuming the generalized continuum hypothesis, we completely determine the cardinals \(\kappa \leq \lambda \) for which a \(\kappa \)-existentially closed group of cardinality \(\lambda \) exists.

MSC:

20B07 General theory for infinite permutation groups
20B35 Subgroups of symmetric groups
03E75 Applications of set theory
20B30 Symmetric groups
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References:

[1] Jech, T.: Set Theory, The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin (2003)
[2] Neumann, BH, The isomorphism problem for algebraically closed groups, Stud. Log. Found. Math., 71, 553-562, (1973) · Zbl 0198.34103
[3] Kegel, OH; Kuzucuoğlu, M, \(κ \)-existentially closed groups, J. Algebra, 499, 298-310, (2018) · Zbl 1427.20002
[4] Scott, WR, Algebraically closed groups, Proc. Am. Math. Soc., 2, 118-121, (1951) · Zbl 0043.02302
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