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Deeply concatenable subgroups might never be free. (English) Zbl 1458.20047

Summary: We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer-Specker group \(\mathbb{Z}^{\omega}\) and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented.

MSC:

20K20 Torsion-free groups, infinite rank
03E25 Axiom of choice and related propositions
03E35 Consistency and independence results
03E75 Applications of set theory
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References:

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