On height sequences in primary Abelian groups

Abstract

For an abelian p-group G, the cyclic subgroups X and Y are defined to be quotient-Ulm equivalent if the quotients G/X and G/Y have the same Ulm function. A complete characterization is given for when X and Y are factor-Ulm equivalent in terms of their respective height sequences. For groups that are quotient-transitive or CS-transitive, this is used to unify, generalize and simplify work of Goldsmith, Gong and Strüngmann (Quotient-Transitivity and Cyclic Subgroup-Transitivity, in J. Group Theory, and Cyclic Subgroup Transitivity for Abelian Groups, to appear in Rend. Semin. Mat. Univ. Padova). These results pertain to the class of transitive groups that satisfy the property satisfy the property that G/X and G/Y are isomorphic whenever they have the same Ulm function. This extensive class contains both the separable groups, as well as the totally projective groups.