Generalized Wallace theorems

Danchev, Peter V.; Keef, Patrick

doi:10.7146/math.scand.a-15083

Cited by 15 publications

(12 citation statements)

References 18 publications

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“…By [19], in the presence of MA + ¬CH, for groups of cardinality ℵ 1 , the classes of weakly ω 1 -separable and ω 1 -separable groups coincide. Finally, in view of Corollary 5.2 of [6] we have that G is weakly ω 1 -separable iff A is weakly ω 1 -separable. Regarding (b), Megibben [19,Theorem 3.2] found in the presence of V = L an ω 1 -separable group A of cardinality ℵ 1 containing a pure and dense subgroup G ′ which is not ω 1 -separable such that A/G ′ is countable.…”

Section: Proof: It Is Clear That (A) Implies (B) Suppose Next That (mentioning

confidence: 91%

“…Proof: Sufficiency follows immediately from Proposition 2.1, and necessity follows directly from Proposition 1.1 of [6] (the second statement does not require the niceness of N in G).…”

Section: Totally Projective Groups and Generalizationsmentioning

confidence: 97%

“…Referring to [6], a group homomorphism ϕ : G → A is said to be ω 1 -bijective if both the kernel and the co-kernel of ϕ are countable. Thus, if there exists an ω 1 -bijective homomorphism ϕ : G → A, then G/ ker ϕ ∼ = ϕ(G) and G is an ℵ 0 -elongation of ϕ(G) by ker ϕ.…”

Section: ])mentioning

confidence: 99%

“…In Example 2.3 of [6], a separable group G was constructed which was not Σ-cyclic but had a (pure) countable subgroup X such that A = G/X was a dsc-group of length ω + 1. It follows that this A is summable, but G is not; in fact G is p ω+1 -projective.…”

Section: N-summable Groupsmentioning

confidence: 99%

“…If we start with the group A from Example 4.6 and construct the group G as in Proposition 4.5, then since G does not have a nice basis, it follows from [2] that it is not p ω+1 -projective. Therefore, in Theorem 4.2 of [6], the condition of separability is necessary and cannot be eliminated.…”

Section: Groups With Nice Basesmentioning

confidence: 99%

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Nice elongations of primary abelian groups

Danchev¹,

Keef²

2010

Publicacions Matemàtiques

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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.

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Section: Proof: It Is Clear That (A) Implies (B) Suppose Next That (mentioning

confidence: 91%

“…Proof: Sufficiency follows immediately from Proposition 2.1, and necessity follows directly from Proposition 1.1 of [6] (the second statement does not require the niceness of N in G).…”

Section: Totally Projective Groups and Generalizationsmentioning

confidence: 97%

Section: ])mentioning

confidence: 99%

Section: N-summable Groupsmentioning

confidence: 99%

Section: Groups With Nice Basesmentioning

confidence: 99%

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Nice elongations of primary abelian groups

Danchev¹,

Keef²

2010

Publicacions Matemàtiques

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

Danchev

Keef

2010

Mediterr. J. Math.

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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.

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On elongations of totally projective groups and α-Σ-groups

Danchev

2009

Lith Math J

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Generalized Wallace theorems

Abstract: We present a number of generalizations of a classical result of Wallace regarding countable extensions of totally projective primary abelian groups.

Cited by 15 publications

References 18 publications

Nice elongations of primary abelian groups

Nice elongations of primary abelian groups

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

On elongations of totally projective groups and α-Σ-groups

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