Infinite Rank Butler Groups

Dugas, Manfred; Rangaswamy, Kulumani M.

doi:10.2307/2001044

Cited by 3 publications

(3 citation statements)

References 4 publications

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“…Lemma 3 [DHR,Proposition 3.9]. In the TEP sequence (I), if both H and G are B2-groups, then so is C, provided C is finitely Butler.…”

Section: Resultsmentioning

confidence: 99%

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A homological characterization of abelian 𝐵₂-groups

Rangaswamy¹

1994

Proc. Amer. Math. Soc.

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“…Lemma 3 [DHR,Proposition 3.9]. In the TEP sequence (I), if both H and G are B2-groups, then so is C, provided C is finitely Butler.…”

Section: Resultsmentioning

confidence: 99%

“…We apply induction on the cardinality k of H. If k < Nj , then replacing C, if necessary, by a suitable C(S) G C , we may assume that \C\ = k and then H is a 52-group by [DHR,Proposition 3.11]. Assume k > Ni and that the theorem holds for cardinalities < k. Suppose k is regular.…”

Section: Theorem 9 (Ch) Suppose 0->//->c->g-+0mentioning

confidence: 99%

A homological characterization of abelian 𝐵₂-groups

Rangaswamy¹

1994

Proc. Amer. Math. Soc.

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“…Various authors have constructed algebraic structures having prescribed endomorphism monoids (including the case of small automorphism groups); see, for instance, [4] for a survey on this problem for modules and fields. In [3], strict partial orders with large endomorphism monoid but small automorphism groups were used to solve a problem from universal algebra on Krasner clones.…”

Section: Introductionmentioning

confidence: 99%

Rigid chains admitting many embeddings

Droste

Truss

2000

Proc. Amer. Math. Soc.

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Abstract. A chain (linearly ordered set) is rigid if it has no non-trivial automorphisms. The construction of dense rigid chains was carried out by Dushnik and Miller for subsets of R, and there is a rather different construction of dense rigid chains of cardinality κ, an uncountable regular cardinal, using stationary sets as 'codes', which was adapted by Droste to show the existence of rigid measurable spaces. Here we examine the possibility that, nevertheless, there could be many order-embeddings of the chain, in the sense that the whole chain can be embedded into any interval. In the case of subsets of R, an argument involving Baire category is used to modify the original one. For uncountable regular cardinals, a more complicated version of the corresponding argument is used, in which the stationary sets are replaced by sequences of stationary sets, and the chain is built up using a tree. The construction is also adapted to the case of singular cardinals.

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Butler groups of infinite rank. II

Dugas¹,

Hill²,

Rangaswamy³

1990

Trans. Amer. Math. Soc.

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A torsion-free abelian group G is called a Butler group if Bext(G, 7") = 0 for any torsion group T . We show that every Butler group G of cardinality N, is a ¿?2"8rouP; 'e-' C7 is a union of a smooth ascending chain of pure subgroups Ga where Ga+X = Ga + Ba , Ba a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding N^ isa ß2-group. Moreover, we are able to prove that the derived functor Bext (A , T) = 0 for any torsion group T and any torsion-free A with \A\ < Kw . This implies that under CH all balanced subgroups of completely decomposable groups of cardinality < Hm are 52"ßrouPs-(1) Bext(t7, T) = 0 for all torsion groups T.(2) Bext(C7, T) = 0 for all 1,-cyclic torsion groups T.(3) G is a B2-group.(ß) The derived function Bext (G, T) = 0 for any torsion group T.

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Infinite Rank Butler Groups

Cited by 3 publications

References 4 publications

A homological characterization of abelian 𝐵₂-groups

A homological characterization of abelian 𝐵₂-groups

Rigid chains admitting many embeddings

Butler groups of infinite rank. II

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