zbMATH — the first resource for mathematics

Weak periodicity and minimality of topological groups. (English) Zbl 0637.22001
The author introduces the notion of a weakly periodic element for every topological Hausdorff group. The set wtd(G) of all weakly periodic elements of a topological group G forms a subgroup. If the group G is compact then the subgroup wtd(G) is totally minimal (a topological group A is called totally minimal if every continuous homomorphism onto another topological group is open). The main result of the paper is the following. Among the classes of minimal abelian groups, which contain the class of compact abelian groups and are closed under products and quotients there exists a greatest. Precompact abelian groups G, such that for every cardinal number \(\tau\) the group \(G^{\tau}\) is minimal, are described.
One can see further results in this direction in the articles of I. R. Prodanov, L. N. Stoyanov [C. R. Acad. Bulg. Sci. 37, 23-26 (1984; Zbl 0546.22001)] and S. Dierolf, U. Schwanengel [Pac. J. Math. 82, 349-355 (1979; Zbl 0388.22002)].
Reviewer: M.I.Ursul

22A05 Structure of general topological groups
54D30 Compactness
22D05 General properties and structure of locally compact groups