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**On countable extensions of primary Abelian groups.**
*(English)*
Zbl 1156.20044

It is well known that every reduced Abelian \(p\)-group that possesses a totally projective subgroup of countable index is itself totally projective. Results of this kind were proved also for some other important classes of Abelian groups, and the paper under review aims to continue this line of research. The author calls an Abelian \(p\)-group \(A\) (strongly) \(p^{\omega+n}\)-totally projective if \(p^\omega A\) is totally projective and \(A/p^\omega A\) is (strongly) \(p^{\omega+n}\)-projective. Analogously, an Abelian \(p\)-group \(A\) is called (strongly) \(p^{\omega+n}\)-summable if \(p^\omega A\) is summable and \(A/p^\omega A\) is (strongly) \(p^{\omega+n}\)-projective.

In this paper, it is proved that for any non-negative integer \(n\), if a reduced Abelian \(p\)-group contains a pure \(p^{\omega+n}\)-totally projective (\(p^{\omega+n}\)-summable) subgroup of countable index, then it is \(p^{\omega+n}\)-totally projective (\(p^{\omega+n}\)-summable) as well. Under the additional assumption that the subgroup is nice, the same is proved for strongly \(p^{\omega+n}\)-totally projective and strongly \(p^{\omega+n}\)-summable groups.

In this paper, it is proved that for any non-negative integer \(n\), if a reduced Abelian \(p\)-group contains a pure \(p^{\omega+n}\)-totally projective (\(p^{\omega+n}\)-summable) subgroup of countable index, then it is \(p^{\omega+n}\)-totally projective (\(p^{\omega+n}\)-summable) as well. Under the additional assumption that the subgroup is nice, the same is proved for strongly \(p^{\omega+n}\)-totally projective and strongly \(p^{\omega+n}\)-summable groups.

Reviewer: Michal Kunc (Brno)

### MSC:

20K10 | Torsion groups, primary groups and generalized primary groups |

20K27 | Subgroups of abelian groups |

20K35 | Extensions of abelian groups |

20K25 | Direct sums, direct products, etc. for abelian groups |