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Butler groups of infinite rank and axiom 3. (English) Zbl 0628.20045

The authors study some aspects of Butler groups of infinite rank and several relations between them. The main results are the following: If a torsionfree group H has a balanced cover for its countable subgroups then it is absolutely separable (Th.2.3). If H is a pure subgroup of a direct sum of countable torsionfree groups, then H has a balanced cover for its countale subgroups (Th.3.3). If H is a pure subgroup of a direct sum G of countable torsionfree groups, then H is separable in G iff it has a balanced cover for its countable subgroups. Moreover, if the cardinality of H does not exceed \(\aleph_ 1\), then these conditions imply that H is a \(B_ 2\)-group (Th.3.7). If H is a pure subgroup of a direct sum G of countable torsionfree groups and if G satisfies the third axiom of countability over H with respect to separable subgroups, then H is a \(B_ 2\)-group. Moreover, in this case, H is the union of a smooth ascending chain \(0=H_ 0\subseteq H_ 1\subseteq...\subseteq H_{\alpha}\subseteq..\). of balanced subgroups of H such that \(H_{\alpha +1}/H_{\alpha}\) is a countable \(B_ 2\)-group for each \(\alpha\) (Th.4.2). A torsionfree group G satisfies the third axiom of countability with respect to descent subgroups, iff it is a \(B_ 2\)- group (Th.5.8). If the torsionfree group G is the union of a smooth ascending chain \(0=G_ 0\subseteq G_ 1\subseteq...\subseteq G_{\alpha}\subseteq..\). of pure and separable subgroups with \(G_{\alpha +1}/G_{\alpha}\) countable for each \(\alpha\), then \(Bext^ 2(G,T)=0\) for every torsion group T (Th.6.3). Bext\({}^ 3(G,T)=0\) for all G torsionfree and all T torsion (Th.6.4).
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
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References:

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