On a class of locally Butler groups

Ladislav Bican

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 597-600
  • ISSN: 0010-2628

Abstract

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A torsionfree abelian group B is called a Butler group if B e x t ( B , T ) = 0 for any torsion group T . It has been shown in [DHR] that under C H any countable pure subgroup of a Butler group of cardinality not exceeding ℵ ω is again Butler. The purpose of this note is to show that this property has any Butler group which can be expressed as a smooth union ∪ α < μ B α of pure subgroups B α having countable typesets.

How to cite

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Bican, Ladislav. "On a class of locally Butler groups." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 597-600. <http://eudml.org/doc/247294>.

@article{Bican1991,
abstract = {A torsionfree abelian group $B$ is called a Butler group if $Bext(B,T) = 0$ for any torsion group $T$. It has been shown in [DHR] that under $CH$ any countable pure subgroup of a Butler group of cardinality not exceeding $\aleph _\omega $ is again Butler. The purpose of this note is to show that this property has any Butler group which can be expressed as a smooth union $\cup _\{\alpha < \mu \}B_\alpha $ of pure subgroups $B_\alpha $ having countable typesets.},
author = {Bican, Ladislav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Butler group; generalized regular subgroup; torsion-free abelian group; Butler group; finite rank pure subgroup; smooth union; ascending chain of pure subgroups; typeset},
language = {eng},
number = {4},
pages = {597-600},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On a class of locally Butler groups},
url = {http://eudml.org/doc/247294},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Bican, Ladislav
TI - On a class of locally Butler groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 597
EP - 600
AB - A torsionfree abelian group $B$ is called a Butler group if $Bext(B,T) = 0$ for any torsion group $T$. It has been shown in [DHR] that under $CH$ any countable pure subgroup of a Butler group of cardinality not exceeding $\aleph _\omega $ is again Butler. The purpose of this note is to show that this property has any Butler group which can be expressed as a smooth union $\cup _{\alpha < \mu }B_\alpha $ of pure subgroups $B_\alpha $ having countable typesets.
LA - eng
KW - Butler group; generalized regular subgroup; torsion-free abelian group; Butler group; finite rank pure subgroup; smooth union; ascending chain of pure subgroups; typeset
UR - http://eudml.org/doc/247294
ER -

References

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  1. Arnold D., Notes on Butler groups and balanced extensions, Boll. Un. Mat. Ital. A(6) 5 (1986), 175-184. (1986) Zbl0601.20050MR0850285
  2. Bican L., Splitting in abelian groups, Czech. Math. J. 28 (1978), 356-364. (1978) Zbl0421.20022MR0480778
  3. Bican L., Purely finitely generated groups, Comment. Math. Univ. Carolinae 21 (1980), 209-218. (1980) MR0580678
  4. Bican L., Salce L., HASH(0x91ad320), Infinite rank Butler groups, Proc. Abelian Group Theory Conference, Honolulu, Lecture Notes in Math., vol. 1006, Springer-Verlag, 1983, 171-189. 
  5. Bican L., Salce L., Štěpán J., A characterization of countable Butler groups, Rend. Sem. Mat. Univ. Padova 74 (1985), 51-58. (1985) MR0818715
  6. Butler M.C.R., A class of torsion-free abelian groups of finite rank, Proc. London Math. Soc. 15 (1965), 680-698. (1965) Zbl0131.02501MR0218446
  7. Dugas M., On some subgroups of infinite rank Butler groups, Rend. Sem. Mat. Univ. Padova 79 (1988), 153-161. (1988) Zbl0667.20043MR0964027
  8. Dugas M., Hill P., Rangaswamy K.M., Infinite rank Butler groups II, Trans. Amer. Math. Soc. 320 (1990), 643-664. (1990) MR0963246
  9. Dugas M., Rangaswamy K.M., Infinite rank Butler groups,, Trans. Amer. Math. Soc. 305 (1988), 129-142. (1988) Zbl0641.20036MR0920150
  10. Fuchs L., Infinite Abelian groups, vol. I and II, Academic Press, New York, 1973 and 1977. Zbl0338.20063MR0255673
  11. Fuchs L., Infinite rank Butler groups, preprint. 
  12. Fuchs L., Metelli C., Countable Butler groups, Contemporary Math., to appear. Zbl0769.20025MR1176115
  13. Fuchs L., Viljoen G., Note on the extensions of Butler groups, Bull. Austral. Math. Soc. 41 (1990), 117-122. (1990) MR1043972

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