On countable extensions of primary abelian groups
Danchev, Peter Vassilev
Archivum Mathematicum, Tome 043 (2007), p. 61-66 / Harvested from Czech Digital Mathematics Library
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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 $p^{\omega +n}$-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.

Publié le : 2007-01-01
Classification:  20K10,  20K15 
@article{108050,
     author = {Peter Vassilev Danchev},
     title = {On countable extensions of primary abelian groups},
     journal = {Archivum Mathematicum},
     volume = {043},
     year = {2007},
     pages = {61-66},
     zbl = {1156.20044},
     mrnumber = {2310125},
     language = {en},
     url = {http://dml.mathdoc.fr/item/108050}
}

Danchev, Peter Vassilev. On countable extensions of primary abelian groups. Archivum Mathematicum, Tome 043 (2007) pp. 61-66. http://gdmltest.u-ga.fr/item/108050/

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) | MR 0276328 | Zbl 0215.39804

Cutler D. Quasi-isomorphism for infinite abelian $p$-groups, Pacific J. Math. 16 (1) (1966), 25–45. (1966) | MR 0191954 | Zbl 0136.28904

Danchev P. Characteristic properties of large subgroups in primary abelian groups, Proc. Indian Acad. Sci. Math. Sci. 104 (3) (2004), 225–233. | MR 2083463 | Zbl 1062.20059

Danchev P. Countable extensions of torsion abelian groups, Arch. Math. (Brno) 41 (3) (2005), 265–272. | MR 2188382 | Zbl 1114.20030

Danchev P. A note on the countable extensions of separable $p^{\omega +n}$-projective abelian $p$-groups, Arch. Math. (Brno) 42 (3) (2006), 251–254. | MR 2260384

Danchev P. Generalized Wallace theorems, submitted. | Zbl 1169.20029

Danchev P. Theorems of the type of Cutler for abelian $p$-groups, submitted. | Zbl 1179.20046

Danchev P. Commutative group algebras of summable $p$-groups, Comm. Algebra 35 (2007). | MR 2313667 | Zbl 1122.20003

Danchev P. Invariant properties of large subgroups in abelian $p$-groups, Oriental J. Math. Sci. 1 (1) (2007). | MR 2656103 | Zbl 1196.20060

Fuchs L. Infinite Abelian Groups, I and II, Mir, Moskva, 1974 and 1977 (in Russian). (1974) | MR 0457533 | Zbl 0338.20063

Fuchs L.; Irwin J. On elongations of totally projective $p$-groups by $p^{\omega +n}$-projective $p$-groups, Czechoslovak Math. J. 32 (4) (1982), 511–515. (1982) | MR 0682128

Nunke R. Homology and direct sums of countable abelian groups, Math. Z. 101 (3) (1967), 182–212. (1967) | MR 0218452 | Zbl 0173.02401

Nunke R. Uniquely elongating modules, Symposia Math. 13 (1974), 315–330. (1974) | MR 0364491 | Zbl 0338.20018

Wallace K. On mixed groups of torsion-free rank one with totally projective primary components, J. Algebra 17 (4) (1971), 482–488. (1971) | MR 0272891 | Zbl 0215.39902