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Cardinalities of minimal Abelian groups. (English) Zbl 0571.22003
Mathematics and education in mathematics, Proc. 10th Spring Conf. Union. Bulg. Math., Sunny Beach 1981, 203-208 (1981).
For the entire collection see Zbl 0484.00003.
Here the author solves the following problem for all free abelian groups: Describe the abelian groups that admit minimal group topologies.
Theorem. (a) If \(G\) is a minimal abelian group, then \(|G|\) is a Stoyanov cardinal, i.e. an infinite cardinal number \(\kappa\) satisfying \(\min(\kappa, \rho)\) for some infinite cardinal \(\rho\) (called “permissible cardinals” by the author). (b) For a cardinal \(\kappa\), \(F_\kappa\) admits minimal group topologies if and only if \(\kappa\) is a Stoyanov cardinal, where \(F_\kappa\) denotes the free abelian group with \(\kappa\) many generators.

MSC:
22A05 Structure of general topological groups