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

Arnold, David M.

Boll. Unione Mat. Ital., VI. Ser., A 5, 175-184 (1986).

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

Keywords:

balanced extension; torsionfree abelian group; chain of pure subgroups; Butler group; B-group; locally completely decomposable; generalized regular subgroup; locally *-Noetherian; direct sum of countable groups; homogeneous torsionfree group; type