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Infinite rank Butler groups. (English) Zbl 0641.20036

A torsionfree Abelian group G is said to be Butler if \(Bext(G,T)=0\) for all torsion groups T. Every finite rank Butler group G has T.E.P. (torsion extension property), i.e. every homomorphism of a pure subgroup of G extends to one of G (Th.2). A torsionfree group G has T.E.P. iff every homomorphic image of G splits (factor-splitting groups). For the study of \(B_ 2\)-groups (the subclass of Butler groups defined by some smooth chains of pure subgroups) the notion of a decent subgroup is useful (Prop. 5). A pure subgroup A of a countable Butler group B is decent iff B has T.E.P. over A (Th. 7). The last part of the paper is devoted to Butler groups of uncountable rank. The main result asserts that under \((V=L)\) a Butler group B of rank \(\aleph_ 1\) is a \(B_ 2\)- group whenever every finite rank pure subgroup of B is Butler.
Reviewer: L.Bican

MSC:

20K20 Torsion-free groups, infinite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K27 Subgroups of abelian groups
20K35 Extensions of abelian groups
20K40 Homological and categorical methods for abelian groups
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