# zbMATH — the first resource for mathematics

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