LAUSR

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