Relative categoricity in abelian groups II

We consider structures A consisting of an abelian group with a subgroup A P 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 ∣ A P ∣ = κ , and given any two such models A , B with A P = B P , there is an isomorphism from A to B which is the identity on A P . 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 A P of order κ which are ( κ , λ )-categorical.