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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References:

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