The existence of certain pure dense subgroups of abelian p-groups is not decidable in ZFC

Abstract

Let κ be a regular uncountable cardinal and n a positive integer. Let G be a separable p-group, and let H be a proper pure dense subgroup of G with a κ-filtration H =∪ αϵκH α, such that {αϵκ: p ω + n( H H α )≠ 0 } is stationary in κ. Assuming V = L, there are 2 κ pairwise nonisomorphic pure subgroups of G each having the same p n -socle as H. The following two questions are not decidable in ZFC. Let H be a pure dense subgroup of a separable p-group G, and let n be a positive integer. Does there exist a pure subgroup K of G such that K[ p n ] = H[ p n ] and K is not isomorphic to H? If A is an abelian p-group, are all high subgroups of A isomorphic?