Deeply concatenable subgroups might never be free
Samuel M. Corson, Saharon Shelah
TL;DR
This paper investigates algebraic structures that are not free without the axiom of choice, focusing on specific subgroups and their implications for various mathematical properties.
Contribution
It demonstrates that certain subgroups, such as those of the Baer-Specker group and Hawaiian earring group, are not free without the axiom of choice, revealing new insights into their structure.
Findings
Some subgroups of $ ext{Z}^ ext{omega}$ are not free without choice
Hawaiian earring group lacks freeness without choice
Applications to slenderness and topological groups
Abstract
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 and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented
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