Nice elongations of primary abelian groups

Abstract

Suppose N is a nice subgroup of the primary abelian group G and A = G/N . The paper discusses various contexts in which G satisfying some property implies that A also satisfies the property, or visa versa, especially when N is countable. For example, if n is a positive integer, G has length not exceeding ω 1 and N is countable, then G is n-summable iff A is n-summable. When A is separable and N is countable, we discuss the condition that any such G decomposes into the direct sum of a countable and a separable group, and we show that it is undecidable in ZFC whether this condition implies that A must be a direct sum of cyclics. We also relate these considerations to the study of nice bases for primary abelian groups. Introduction and TerminologyBy the term "group" we will mean an abelian p-group, where p is a prime fixed for the duration. Our group theoretic terminology and notation will generally follow [11]. We will say a group G is Σ-cyclic if it is isomorphic to a direct sum of cyclic groups. We will also utilize the language of valuated groups and valuated vector spaces (see [22] and [12], respectively); for example, if Y is a valuated group, the letter v will be reserved for its valuation, and for any ordinal α, by Y (α) we will mean the subgroup {y ∈ Y : v(y) ≥ α}. We will implicitly assume that all valuated vector spaces are over Z p .If 0 → X → G → A → 0 is a short exact sequence, we will routinely identify X with an actual subgroup of G and A with the quotient group G/X; we will say that G is an elongation of A by X. We then say that G is a nice-elongation of A if X is a nice subgroup (i.e., for every y ∈ G, the coset y + X has an element of maximum height). Further, 2000 Mathematics Subject Classification. 20K10.

ON ALMOST ω1-pω+n-PROJECTIVE ABELIAN p-GROUPS

“…Theorem 1.7. ( [5]) Let f : G → H be an ω 1 -bijection such that both G and H are reduced groups. If G is almost totally projective, then H is almost totally projective.…”

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

“…A κ-coseparable valuated vector space will be said to be proper if it is not free. In [6] the existence of a proper ℵ 1 -coseparable valuated vector space was shown to be equivalent to a question involving the structure of abelian groups, and to be independent of ZFC. We conclude this paper by showing that for 0 < n < ω, the existence of a proper ℵ 0 -coseparable valuated vector space is equivalent to the existence of a proper ω + n-totally p ω+n -projective group, and we prove that both of these propositions are independent of ZFC (Theorem 3.11).…”

On almost ω1-weak p ω·2+n -projective abelian p-groups

“…(e) Let : → be an 1 -bijection such that both and are separable groups. If is almost Σ-cyclic, then is almost Σ-cyclic [1,5]. A few useful assertions follow here.…”

“…(5) is an extension of a -bounded group by an almost 1 -weak ⋅2 -projective factor. (6) is an extension of a -countable group by an almost weak ⋅2 -projective factor.…”