Countable extensions of torsion Abelian groups

Peter Vassilev Danchev

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 3, page 265-272
  • ISSN: 0044-8753

Abstract

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Suppose A is an abelian torsion group with a subgroup G such that A / G is countable that is, in other words, A is a torsion countable abelian extension of G . A problem of some group-theoretic interest is that of whether G ∈ 𝕂 , a class of abelian groups, does imply that A ∈ 𝕂 . The aim of the present paper is to settle the question for certain kinds of groups, thus extending a classical result due to Wallace (J. Algebra, 1981) proved when 𝕂 coincides with the class of all totally projective p -groups.

How to cite

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Danchev, Peter Vassilev. "Countable extensions of torsion Abelian groups." Archivum Mathematicum 041.3 (2005): 265-272. <http://eudml.org/doc/249484>.

@article{Danchev2005,
abstract = {Suppose $A$ is an abelian torsion group with a subgroup $G$ such that $A/G$ is countable that is, in other words, $A$ is a torsion countable abelian extension of $G$. A problem of some group-theoretic interest is that of whether $G \in \mathbb \{K\}$, a class of abelian groups, does imply that $A\in \mathbb \{K\}$. The aim of the present paper is to settle the question for certain kinds of groups, thus extending a classical result due to Wallace (J. Algebra, 1981) proved when $\mathbb \{K\}$ coincides with the class of all totally projective $p$-groups.},
author = {Danchev, Peter Vassilev},
journal = {Archivum Mathematicum},
keywords = {countable factor-groups; $\Sigma $-groups; $\sigma $-summable groups; summable groups; $p^\{\omega + n\}$-projective groups; countable factor-groups; -groups; summable groups; Abelian torsion groups; torsion Abelian extensions; totally projective -groups},
language = {eng},
number = {3},
pages = {265-272},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Countable extensions of torsion Abelian groups},
url = {http://eudml.org/doc/249484},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Danchev, Peter Vassilev
TI - Countable extensions of torsion Abelian groups
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 3
SP - 265
EP - 272
AB - Suppose $A$ is an abelian torsion group with a subgroup $G$ such that $A/G$ is countable that is, in other words, $A$ is a torsion countable abelian extension of $G$. A problem of some group-theoretic interest is that of whether $G \in \mathbb {K}$, a class of abelian groups, does imply that $A\in \mathbb {K}$. The aim of the present paper is to settle the question for certain kinds of groups, thus extending a classical result due to Wallace (J. Algebra, 1981) proved when $\mathbb {K}$ coincides with the class of all totally projective $p$-groups.
LA - eng
KW - countable factor-groups; $\Sigma $-groups; $\sigma $-summable groups; summable groups; $p^{\omega + n}$-projective groups; countable factor-groups; -groups; summable groups; Abelian torsion groups; torsion Abelian extensions; totally projective -groups
UR - http://eudml.org/doc/249484
ER -

References

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  2. Danchev P. V., Commutative group algebras of abelian Σ -groups, Math. J. Okayama Univ. 40 (1998), 77–90. (1998) MR1755921
  3. Danchev P. V., Commutative group algebras of highly torsion-complete abelian p -groups, Comment. Math. Univ. Carolin. (4) 44 (2003), 587–592. Zbl1101.20001MR2062875
  4. Danchev P. V., Commutative group algebras of summable abelian p -groups, Comm. Algebra, in press. 
  5. Danchev P. V., Generalized Dieudonné criterion, Acta Math. Univ. Comenian. (1) 74 (2005), 15–24. Zbl1111.20045MR2154393
  6. Fuchs L., Infinite Abelian Groups, Volumes I and II, Mir, Moskva, 1974 and 1977. (In Russian.) (1974) Zbl0338.20063MR0457533
  7. Hill P. D., Criteria for freeness in groups and valuated vector spaces, Lect. Notes in Math. 616 (1977), 140–157. (1977) Zbl0372.20041MR0486206
  8. Hill P. D., Megibben C. K., On direct sums of countable groups and generalizations, Études sur les Groupes Abéliens, Paris (1968), 183–206. (1968) Zbl0203.32705MR0242943
  9. Hill P. D., Megibben C. K., Extending automorphisms and lifting decompositions in abelian groups, Math. Annalen 175 (1968), 159–168. (1968) Zbl0183.03202MR0223449
  10. Megibben C. K., The generalized Kulikov criterion, Canad. J. Math. 21 (1969), 1192–1205. (1969) Zbl0208.03502MR0249509
  11. Megibben C. K., Countable extensions of simply presented groups, Internet information. 
  12. Wallace K. D., On mixed groups of torsion-free rank one with totally projective primary components, J. Algebra 17 (1971), 482–488. (1971) Zbl0215.39902MR0272891
  13. Nunke R. J., Purity and subfunctors of the identity, Topics in Abelian Groups, Scott Foresman and Co., (1963), 121–171. (1963) MR0169913

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