Relative categoricity in abelian groups II

We consider structures A consisting of an abelian group with a subgroup AP distinguished by a 1-ary relation symbol P, and complete theories T of such structures. Such a theory T is (κ,λ)-categorical if T has models A of cardinality λ with ∣AP∣=κ, and given any two such models A,B with AP=BP, there is an isomorphism from A to B which is the identity on AP. We classify all complete theories of such structures A in terms of the cardinal pairs (κ,λ) in which they are categorical. We classify algebraically the A of finite order λ with AP of order κ which are (κ,λ)-categorical.