On the existence of kappa-existentially closed groups

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