Butler groups of infinite rank. (English) Zbl 0812.20032

Bican, Ladislav

Czech. Math. J. 44, No. 1, 67-79 (1994).

A torsion-free abelian group \(G\) is called a \(B_ 1\)-group if, for any torsion group \(T\), every balanced extension of \(T\) by \(G\) splits. \(G\) is called \(B_ 2\)-group if \(G\) is the union of a continuous well-ordered ascending chain \(0 = G_ 0 \subset \dots \subset G_ \alpha \subset G_{\alpha + 1} \dots\) \((\alpha < \tau)\) of pure subgroups where \(\tau\) is some fixed ordinal and, for each \(\alpha < \tau\), \(G_{\alpha + 1} = G_ \alpha + B_ \alpha\) with \(B_ \alpha\) a pure subgroup of a finite rank completely decomposable group. A \(B_ 2\)-group \(G\) is always a \(B_ 1\)-group and the converse holds if \(| G| \leq \aleph_ 1\). When \(| G| > \aleph_ 1\), additional set-theoretical hypotheses are needed to ensure the existence in \(G\) of a smooth ascending chain of pure separative (separable in Hill’s sense) subgroups connecting 0 to \(G\) with successive factors countable. Then the machinery developed by M. Dugas, P. Hill and the reviewer [Trans. Am. Math. Soc. 320, 643-664 (1990; Zbl 0708.20018)] can be used to show that a \(B_ 1\)-group \(G\) with such a chain is indeed a \(B_ 2\)-group. Dugas, Hill and the reviewer (loc. cit.) showed that the Continuum Hypothesis will yield the needed separative chain if \(| G| \leq \aleph_ \omega\) and L. Fuchs and M. Magidor [Isr. J. Math. 84, 239-263 (1993; Zbl 0789.20062)] showed that, under \(V = L\), groups of arbitrary cardinality will have such a separative chain. In this note, instead of assuming a set-theoretical hypothesis, the author assumes three “algebraic” hypotheses (EPS), (EPR) and (\(\text{EB}_ 2\)) for a \(B_ 1\)-group \(G\) and shows that \(G\) must then be a \(B_ 2\)-group. These three hypotheses, in particular, state that \(G\) has a smooth ascending chain of pure subgroups which are pre-separative – a condition weaker than separativity and further require that whenever \(G/H\) is a \(B_ 1\)-group for a prebalanced subgroup \(H\), then every prebalanced subgroup of smaller cardinality in \(G/H\) must be a \(B_ 2\)-group. {Reviewer’s remark: In a forthcoming article in J. Pure Appl. Algebra, L. Fuchs shows that a \(B_ 1\)-group \(G\) is \(B_ 2\)-group if and only if \(G\) has a smooth preseparative chain from 0 to \(G\) with successive factors countable}.

Reviewer: K.M.Rangaswamy (Colorado Springs)

MSC:

20K20	Torsion-free groups, infinite rank
20K27	Subgroups of abelian groups
03C50	Models with special properties (saturated, rigid, etc.)

Keywords:

Butler groups; separative subgroups; \(B_ 1\)-groups; \(B_ 2\)-groups; Continuum Hypothesis; \(V=L\); balanced extension; continuous well-ordered ascending chain; pure subgroups; finite rank completely decomposable group; separative chain; smooth ascending chain of pure subgroups; prebalanced subgroup; smooth preseparative chain

Citations:

Zbl 0708.20018; Zbl 0789.20062

References:

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