κ-Existentially closed groups

Let κ be an infinite cardinal. The class of κ-existentially closed groups is defined and their basic properties are studied. Moreover, for an uncountable cardinal κ, uniqueness of κ-existentially closed groups are shown, provided that they exist. We also show that for each regular strong limit cardinal κ, there exists κ-existentially closed groups. The structure of centralizers of subgroups of order less than κ in a κ-existentially group G are determined up to isomorphism namely, for any subgroup F≤Gν in G with |F|<κ, the subgroup CG(F) is isomorphic to an extension of Z(F) by G.