Almost coproducts of finite cyclic groups. (English) Zbl 0845.20038

Hill, Paul

Commentat. Math. Univ. Carol. 36, No. 4, 795-804 (1995).

The ambient space is the class of \(p\)-primary abelian groups, Hausdorff in the \(p\)-adic topology. A group \(G\) from such a class is called almost a coproduct of cyclic groups, if it has a weak axiom 3 system of closed subgroups. \(\mathcal C\) is such a system if (1) \(0\in{\mathcal C}\), (2) The union of an ascending chain of subgroups belonging to \(\mathcal C\) again belongs to \(\mathcal C\), (3) For every countable subgroup \(B\) of \(G\), there exists a countable subgroup in \(\mathcal C\) containing \(B\). Some results are as follows: Theorem 3. Let \(G\) be a Hausdorff \(p\)-group of cardinality not exceeding \(\aleph_1\). Then \(G\) is a coproduct of cyclic groups if and only if \(G\) is the union of a smooth ascending chain of pure and closed subgroups that are themselves coproducts of cyclic groups. A number of other sufficient conditions are given for a group to be almost a coproduct of cyclic groups. A coproduct of groups that are almost coproducts of cyclic groups is again almost a coproduct of cyclic groups. The paper ends with an open problem: If \(G\) is almost a coproduct of cyclic groups, does a summand of \(G\) have to be almost a coproduct of cyclic groups? This is a well-written paper, as many other of Hill’s papers.

Reviewer: R.Dimitrić (Berkeley)

MSC:

20K10	Torsion groups, primary groups and generalized primary groups
20K25	Direct sums, direct products, etc. for abelian groups
20K45	Topological methods for abelian groups

Keywords:

\(p\)-primary Abelian groups; \(p\)-adic topology; weak axiom 3 systems of closed subgroups; ascending chain of subgroups; Hausdorff \(p\)-groups; almost coproducts of cyclic groups; summands