## 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.
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