Geodesic
Countable extensions of torsion Abelian groups
Archivum mathematicum, Tome 41 (2005) no. 3, pp. 265-272.

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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 $G \in \mathbb K$, 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.
Classification : 20K10, 20K35
Keywords: countable factor-groups; $\Sigma $-groups; $\sigma $-summable groups; summable groups; $p^{\omega + n}$-projective groups 
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     author = {Danchev, Peter},
     title = {Countable extensions of torsion {Abelian} groups},
     journal = {Archivum mathematicum},
     pages = {265--272},
     publisher = {mathdoc},
     volume = {41},
     number = {3},
     year = {2005},
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     zbl = {1114.20030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2005__41_3_a2/}
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Danchev, Peter. Countable extensions of torsion Abelian groups. Archivum mathematicum, Tome 41 (2005) no. 3, pp. 265-272. http://geodesic.mathdoc.fr/item/ARM_2005__41_3_a2/