An Application of Set Theory to ω + n-Totally p ω+n -Projective Primary Abelian Groups

Abstract

Several equivalent descriptions are given of the class of primary abelian groups whose separable subgroups are all direct sums of cyclic groups; such groups are called ω-totally Σ-cyclic. This establishes the converse of a theorem due to Megibben. For n < ω, this is generalized to a consideration of the class of primary abelian groups whose p ω+n -bounded subgroups are all p ω+n -projective. The question of whether there are such groups that are proper in the sense that they are neither p ω+n -projective nor ω-totally Σ-cyclic is shown to be logically equivalent to a natural question about the structure of valuated vector spaces. Finally, it is shown that both of these statements are independent of ZFC.Mathematics Subject Classification (2010). Primary 20K10.

Realization Theorems for Valuated $p^n$-Socles

“…With this cardinality assumption, it is shown that V is uniquely realizable iff V (v n À 1) is countable; and in this case, every group G supported by V will be a dsc group. In particular, when n 1, Corollary 3.5 states that these groups agree with those described in ( [2], Theorem 2.6). Theorem 2.11 (i.e., the solution to the above question (2)) is a generalization of the classical``Existence Theorem for Principal p-Groups'' from [9]; in fact, for n 1, it reduces to precisely this result.…”

“…These groups were studied in [2], where, for example, it was shown that they are precisely the dsc groups G for which p v G is countable. Therefore, we have the following consequence of Theorem 3.4 for n 1.…”

“…This combinatorial condition, which we describe later, can be viewed as a generalization of the classical notion of an admissible function (see [6], Theorem 83.6). Section 2 is a consideration of question (2). In Theorem 2.11 it is shown that V is realizable iff for every countable limit ordinal l and every a5l we have lnÀ1 b5lv…”

On properties of n-totally projective Abelian p-groups

“…Note that these groups G are classified in [4] up to an isomorphism using their Ulm invariants, together with the Ulm invariants of the totally projective groups E λ G, over all limit ordinals λ of uncountable cofinality. Next, since a direct sum of groups is (strongly) n-totally projective if and only if each of its terms has that property, and since the functor E λ G also respects direct sums (because λ has uncountable cofinality), we may assume that G is a λ-elementary A-group and that we possess a representing sequence as in (1). Notice that for any limit ordinal β < λ, we have a balanced-exact sequence implied via (1)…”

“…The next example shows that the class of strongly n-totally projective groups is not contained in the class of ω + n-totally p ω+n -projective groups. Recall that in [1] a group G is said to be ω + n-totally p ω+n -projective group if each p ω+n -bounded subgroup is p ω+n -projective. Example 2.…”