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**Notes on Butler groups and balanced extensions.**
*(English)*
Zbl 0601.20050

Recall, that a torsionfree abelian group G is said to be a B-group if \(Bext(G,T)=0\) for each torsion group T. Consider the following conditions: (i) G is the union of a smooth chain of pure subgroups \(0=G_ 0\subset G_ 1\subset...\subset G_{\lambda}\subset G_{\lambda +1}\subset...\subset G\) such that for each \(\lambda\) there is a Butler group \(B_{\lambda}\) with \(G_{\lambda +1}=G_{\lambda}+B_{\lambda}\); (ii) G is a B-group; (iii) G is locally completely decomposable and if K is a generalized regular subgroup of G and H is a pure finite rank subgroup of G, then \((H/H\cap K)_ p=0\) for almost all primes p; (iv) G is locally completely decomposable and locally *-Noetherian; (v) G is locally completely decomposable and each pure finite rank subgroup of G is a Butler group.

The author presents the proofs of the following implications: (i)\(\Rightarrow (ii)\Rightarrow (iii)\Leftrightarrow (iv)\Leftarrow (v)\Leftarrow (i)\). If G is a direct sum of countable groups, then all these conditions are equivalent. If G is a homogeneous torsionfree group with idempotent type, then G is a B-group iff it is completely decomposable. It is noted that it is known that the hypothesis that G has idempotent type can be omitted.

The author presents the proofs of the following implications: (i)\(\Rightarrow (ii)\Rightarrow (iii)\Leftrightarrow (iv)\Leftarrow (v)\Leftarrow (i)\). If G is a direct sum of countable groups, then all these conditions are equivalent. If G is a homogeneous torsionfree group with idempotent type, then G is a B-group iff it is completely decomposable. It is noted that it is known that the hypothesis that G has idempotent type can be omitted.

Reviewer: L.Bican

### MSC:

20K20 | Torsion-free groups, infinite rank |

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

20K27 | Subgroups of abelian groups |

20K35 | Extensions of abelian groups |