On countable extensions of primary abelian groups

Peter Vassilev Danchev

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 1, page 61-66
  • ISSN: 0044-8753

Abstract

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It is proved that if A is an abelian p -group with a pure subgroup G so that A / G is at most countable and G is either p ω + n -totally projective or p ω + n -summable, then A is either p ω + n -totally projective or p ω + n -summable as well. Moreover, if in addition G is nice in A , then G being either strongly p ω + n -totally projective or strongly p ω + n -summable implies that so is A . This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective p -groups as well as continues our recent investigations in (Arch. Math. (Brno), 2005 and 2006). Some other related results are also established.

How to cite

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Danchev, Peter Vassilev. "On countable extensions of primary abelian groups." Archivum Mathematicum 043.1 (2007): 61-66. <http://eudml.org/doc/250162>.

@article{Danchev2007,
abstract = {It is proved that if $A$ is an abelian $p$-group with a pure subgroup $G$ so that $A/G$ is at most countable and $G$ is either $p^\{\omega +n\}$-totally projective or $p^\{\omega +n\}$-summable, then $A$ is either $p^\{\omega +n\}$-totally projective or $p^\{\omega +n\}$-summable as well. Moreover, if in addition $G$ is nice in $A$, then $G$ being either strongly $p^\{\omega +n\}$-totally projective or strongly $p^\{\omega +n\}$-summable implies that so is $A$. This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$-groups as well as continues our recent investigations in (Arch. Math. (Brno), 2005 and 2006). Some other related results are also established.},
author = {Danchev, Peter Vassilev},
journal = {Archivum Mathematicum},
keywords = {countable quotient groups; $\omega $-elongations; $p^\{\omega +n\}$-totally projective groups; $p^\{\omega +n\}$-summable groups; countable quotient groups; -elongations; Abelian -groups; pure subgroups; summable groups; totally projective -groups},
language = {eng},
number = {1},
pages = {61-66},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On countable extensions of primary abelian groups},
url = {http://eudml.org/doc/250162},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Danchev, Peter Vassilev
TI - On countable extensions of primary abelian groups
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 1
SP - 61
EP - 66
AB - It is proved that if $A$ is an abelian $p$-group with a pure subgroup $G$ so that $A/G$ is at most countable and $G$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable, then $A$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable as well. Moreover, if in addition $G$ is nice in $A$, then $G$ being either strongly $p^{\omega +n}$-totally projective or strongly $p^{\omega +n}$-summable implies that so is $A$. This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$-groups as well as continues our recent investigations in (Arch. Math. (Brno), 2005 and 2006). Some other related results are also established.
LA - eng
KW - countable quotient groups; $\omega $-elongations; $p^{\omega +n}$-totally projective groups; $p^{\omega +n}$-summable groups; countable quotient groups; -elongations; Abelian -groups; pure subgroups; summable groups; totally projective -groups
UR - http://eudml.org/doc/250162
ER -

References

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  1. Benabdallah K., Eisenstadt B., Irwin J., Poluianov E., The structure of large subgroups of primary abelian groups, Acta Math. Acad. Sci. Hungar. 21 (3-4) (1970), 421–435. (1970) Zbl0215.39804MR0276328
  2. Cutler D., Quasi-isomorphism for infinite abelian p -groups, Pacific J. Math. 16 (1) (1966), 25–45. (1966) Zbl0136.28904MR0191954
  3. Danchev P., Characteristic properties of large subgroups in primary abelian groups, Proc. Indian Acad. Sci. Math. Sci. 104 (3) (2004), 225–233. Zbl1062.20059MR2083463
  4. Danchev P., Countable extensions of torsion abelian groups, Arch. Math. (Brno) 41 (3) (2005), 265–272. Zbl1114.20030MR2188382
  5. Danchev P., A note on the countable extensions of separable p ω + n -projective abelian p -groups, Arch. Math. (Brno) 42 (3) (2006), 251–254. MR2260384
  6. Danchev P., Generalized Wallace theorems, submitted. Zbl1169.20029
  7. Danchev P., Theorems of the type of Cutler for abelian p -groups, submitted. Zbl1179.20046
  8. Danchev P., Commutative group algebras of summable p -groups, Comm. Algebra 35 (2007). Zbl1122.20003MR2313667
  9. Danchev P., Invariant properties of large subgroups in abelian p -groups, Oriental J. Math. Sci. 1 (1) (2007). Zbl1196.20060MR2656103
  10. Fuchs L., Infinite Abelian Groups, I and II, Mir, Moskva, 1974 and 1977 (in Russian). (1974) Zbl0338.20063MR0457533
  11. Fuchs L., Irwin J., On elongations of totally projective p -groups by p ω + n -projective p -groups, Czechoslovak Math. J. 32 (4) (1982), 511–515. (1982) MR0682128
  12. Nunke R., Homology and direct sums of countable abelian groups, Math. Z. 101 (3) (1967), 182–212. (1967) Zbl0173.02401MR0218452
  13. Nunke R., Uniquely elongating modules, Symposia Math. 13 (1974), 315–330. (1974) Zbl0338.20018MR0364491
  14. Wallace K., On mixed groups of torsion-free rank one with totally projective primary components, J. Algebra 17 (4) (1971), 482–488. (1971) Zbl0215.39902MR0272891

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