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Butler groups as smooth ascending unions. (English) Zbl 0969.20027

The paper deals with Butler groups of infinite rank and includes the several definitions needed for the subject. The main results are the following. Proposition 4. Let \(\mu\) be an infinite cardinal and let \(H\) be a \(B_2\)-subgroup of a torsion-free group \(G\). If \(|G/H|\leq\mu\), then \(G\) is a \(B_2\)-group if and only if every subset of \(G\) having cardinality at most \(\mu\) embeds in a \(B_2\)-subgroup of cardinality at most \(\mu\). – There are three nice corollaries.
Theorem 10. Let \(\lambda\) be an infinite cardinal and \(G=\bigcup_{\alpha<\lambda} G_\alpha\) be the union of a smooth ascending chain of prebalanced \(B_2\)-subgroups. If \(|G/G_0|=\mu\) and, for each \(\alpha<\lambda\), \(G_{\alpha+1}/G_\alpha\) has a special \(\mu\)-cover of \(B_1\)-subgroups, then \(G\) is a \(B_2\)-group.
Theorem 11. Let \(\lambda\) be an uncountable regular cardinal and \(G=\bigcup_{\alpha<\lambda} G_\alpha\) be the union of a smooth \(\lambda\)-filtration of prebalanced \(B_2\)-subgroups. Then \(G\) is a \(B_2\)-group if and only if there is a cub \(C\subseteq\lambda\) such that \(G_\beta/G_\alpha\) has a special \(\lambda\)-cover of \(B_1\)-subgroups whenever \(\beta\) is the successor of \(\alpha\) in \(C\).

MSC:

20K20 Torsion-free groups, infinite rank
20K27 Subgroups of abelian groups
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