Butler groups and Shelah's Singular Compactness

Ladislav Bican

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 1, page 171-178
  • ISSN: 0010-2628

Abstract

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A torsion-free group is a B 2 -group if and only if it has an axiom-3 family ℭ of decent subgroups such that each member of ℭ has such a family, too. Such a family is called S L ℵ 0 -family. Further, a version of Shelah’s Singular Compactness having a rather simple proof is presented. As a consequence, a short proof of a result [R1] stating that a torsion-free group B in a prebalanced and TEP exact sequence 0 → K → C → B → 0 is a B 2 -group provided K and C are so.

How to cite

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Bican, Ladislav. "Butler groups and Shelah's Singular Compactness." Commentationes Mathematicae Universitatis Carolinae 37.1 (1996): 171-178. <http://eudml.org/doc/247938>.

@article{Bican1996,
abstract = {A torsion-free group is a $B_2$-group if and only if it has an axiom-3 family $\mathfrak \{C\}$ of decent subgroups such that each member of $\mathfrak \{C\}$ has such a family, too. Such a family is called $SL_\{\aleph _0\}$-family. Further, a version of Shelah’s Singular Compactness having a rather simple proof is presented. As a consequence, a short proof of a result [R1] stating that a torsion-free group $B$ in a prebalanced and TEP exact sequence $0 \rightarrow K \rightarrow C \rightarrow B \rightarrow 0$ is a $B_2$-group provided $K$ and $C$ are so.},
author = {Bican, Ladislav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$B_1$-group; $B_2$-group; prebalanced subgroup; torsion extension property; decent subgroup; axiom-3 family; torsion-free Abelian groups; ascending chains of pure subgroups; singular compactness; -groups; singular cardinals; pure -subgroups; infinite rank Butler groups},
language = {eng},
number = {1},
pages = {171-178},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Butler groups and Shelah's Singular Compactness},
url = {http://eudml.org/doc/247938},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Bican, Ladislav
TI - Butler groups and Shelah's Singular Compactness
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 1
SP - 171
EP - 178
AB - A torsion-free group is a $B_2$-group if and only if it has an axiom-3 family $\mathfrak {C}$ of decent subgroups such that each member of $\mathfrak {C}$ has such a family, too. Such a family is called $SL_{\aleph _0}$-family. Further, a version of Shelah’s Singular Compactness having a rather simple proof is presented. As a consequence, a short proof of a result [R1] stating that a torsion-free group $B$ in a prebalanced and TEP exact sequence $0 \rightarrow K \rightarrow C \rightarrow B \rightarrow 0$ is a $B_2$-group provided $K$ and $C$ are so.
LA - eng
KW - $B_1$-group; $B_2$-group; prebalanced subgroup; torsion extension property; decent subgroup; axiom-3 family; torsion-free Abelian groups; ascending chains of pure subgroups; singular compactness; -groups; singular cardinals; pure -subgroups; infinite rank Butler groups
UR - http://eudml.org/doc/247938
ER -

References

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  1. Albrecht U., Hill P., Butler groups of infinite rank and axiom 3, Czech. Math. J. 37 (1987), 293-309. (1987) Zbl0628.20045MR0882600
  2. Bican L., Purely finitely generated groups, Comment. Math. Univ. Carolinae 21 (1980), 209-218. (1980) MR0580678
  3. Bican L., Butler groups of infinite rank, Czech. Math. J. 44 (119) (1994), 67-79. (1994) Zbl0812.20032MR1257937
  4. Bican L., On B 2 -groups, Contemporary Math. 171 (1994), 13-19. (1994) MR1293129
  5. Bican L., Families of preseparative subgroups, to appear. Zbl0866.20043MR1415629
  6. Bican L., Fuchs L., Subgroups of Butler groups, Communications in Algebra 22 (1994), 1037-1047. (1994) Zbl0802.20045MR1261020
  7. Bican L., Salce L., Infinite rank Butler groups, Proc. Abelian Group Theory Conference, Honolulu Lecture Notes in Math., Springer-Verlag 1006 (1983), 171-189. (1983) 
  8. 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
  9. Dugas M., Hill P., Rangaswamy K.M., Infinite rank Butler groups II, Trans. Amer. Math. Soc. 320 (1990), 643-664. (1990) MR0963246
  10. Fuchs L., Infinite Abelian Groups, vol. I and II, Academic Press New York (1973 and 1977). (1973 and 1977) MR0255673
  11. Fuchs L., Infinite rank Butler groups, preprint. 
  12. Hodges W., In singular cardinality, locally free algebras are free, Algebra Universalis 12 (1981), 205-220. (1981) Zbl0476.03039MR0608664
  13. Rangaswamy K.M., A homological characterization of abelian B 2 -groups, Comment. Math. Univ. Carolinae 35 (1994), 627-631. (1994) 
  14. Rangaswamy K.M., A property of B 2 -groups, Proc. Amer. Math. Soc. 121 (1994), 409-415. (1994) MR1186993

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