High subgroups of primary abelian groups of length ω + n

Abstract

Let α be a limit ordinal, n a positive integer and A an abelian p-group of length α + n. We give a characterization of those subgroups of A p αA that are images of p α + i -high subgroups of A. Using this we show that the study of abelian p-groups of length ω + n having all high subgroups isomorphic is equivalent to the study of groups G of length ω having a specified set of pure dense subgroups isomorphic. This set of pure dense subgroups of G is determined by a dense subgroup P of G[ p n ] modulo a maximal p n − 1 -bounded summand of G. For each positive integer n we give an example of a p-group A such that all the high subgroups of A are isomorphic but not all the high subgroups of A p ω + n − 1A are isomorphic.