Simply presented valuated abelian p-groups

Hunter, Roger; Richman, Fred; Walker, Elbert A.

doi:10.1016/0021-8693(77)90272-1

“…But H(ß + 1) = K(ß + 1), so JH(X, X + n) = JK(X, X + n) for all X > 9 and positive integers n. If 9 + u < ß, the same is true by Lemma 9, part (6). Thus (1) is equivalent to (2). That (2) is equivalent to (3) is clear.…”

Section: Valuated Trees and Groupsmentioning

confidence: 88%

“…An element x of a tree X has order n if p"x = 0 and p"~xx ^ 0. This definition of a tree differs from that given in [2] in that we have not restricted ourselves to torsion trees, that is, trees in which every element has finite order. For a tree X and ordinal a, the subset p "AT is defined inductively by setting p°X = X and p"X= C\piPßX) ß«x when a > 0.…”

Section: Valuated Trees and Groupsmentioning

confidence: 99%

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Existence Theorems for Warfield Groups

Hunter¹,

Richman²,

Walker³

1978

Transactions of the American Mathematical Society

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Abstract. Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.

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“…(l)p0 = 0, (2)p"x = x only if n = 0 or x = 0. An element x of a tree X has order n if p"x = 0 and p"~xx ^ 0.…”

Section: Valuated Trees and Groupsmentioning

confidence: 99%

Existence theorems for Warfield groups

Hunter¹,

Richman²,

Walker³

1978

Trans. Amer. Math. Soc.

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Abstract. Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.

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“…The valuated p-local groups are the objects of the category V p studied extensively by Hunter, Richman and Walker (e.g. see [7], [8] and [11]). A group homomorphism α : (G, v) → (H, w) is a V p -morphism if w(α(x)) ≥ v(x) for all x ∈ G, and we write α ∈ Mor(G, H) in this case.…”

Section: Introductionmentioning

confidence: 99%

Finite valuated groups as modules over their endomorphism ring

Albrecht

¹

2017

Stud. Univ. Babes-Bolyai Math.

0

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Abstract. This paper discusses the structure of a finite valuated p-group when viewed as a module over its endomorphism ring. A category equivalence between full subcategories of the category of valuated p-groups and the category of right modules over the endomorphism ring of A is used to investigate the interaction between this module structure and homological properties of the underlying group. Examples are given throughout the paper.

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Simply presented valuated abelian p-groups

Cited by 16 publications

References 4 publications

Existence Theorems for Warfield Groups

Existence Theorems for Warfield Groups

Existence theorems for Warfield groups

Finite valuated groups as modules over their endomorphism ring

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