Classifications of countably-based Abelian profinite groups. (Q2856534)

The author classifies countably based (i.e. second countable) Abelian profinite groups applying Pontryagin duality to the known classification of countable Abelian torsion groups from the book of \textit{L. Fuchs} [Abelian groups. Oxford-London-New York-Paris: Pergamon Press (1960; Zbl 0100.02803)]. The classification is in terms of torsion of the group.NEWLINENEWLINE For a profinite group \(G\) denote by \(t(G)\) the set of torsion elements. Let \(G\) be an Abelian pro-\(p\) group. The author defines the torsion sequence \(T_{\alpha+1}(G)/T_\alpha(G)\) of \(G\) as the group \(\overline{t(G)/T_\alpha(G)}\) for any ordinal number \(\alpha\), where \(T_0(G)=\{1\}\) and overline means topological closure (which is a closed subgroup). If \(\delta\) is a limit ordinal \(T_\delta(G)\) is just the closure of the union \(\bigcup_{\alpha<\delta}T_\alpha(G)\). The ordinal \(\tau\) such that \(T_\tau(G)=T_{\tau+1}(G)\) the author calls the torsion type of \(G\).NEWLINENEWLINE The author proves then that every term of the torsion sequence is a direct (Cartesian) product of cyclic groups and that this is also true for the subgroup generated by torsion.NEWLINENEWLINE An Abelian pro-\(p\) group that does not have non-trivial torsion free quotients is called dual-reduced in the paper. Since any Abelian pro-\(p\) group is a direct product of a free Abelian pro-\(p\) group and a dual reduced pro-\(p\) group, the problem of classification is reduced to the classification of dual-reduced Abelian pro-\(p\) groups. Then the author gives the following characterization.NEWLINENEWLINE 1. Countably based Abelian dual reduced pro-\(p\) groups are isomorphic if and only if they have the same torsion sequences.NEWLINENEWLINE 2. For every torsion sequence there exists an Abelian pro-\(p\) group having it as its torsion sequence.NEWLINENEWLINE Then the author proves that a countably based group with unbounded torsion contains any countably generated Abelian pro-\(p\) subgroup as an abstract subgroup and as a corollary deduces that two countably based Abelian pro-\(p\) groups are abstractly isomorphic if and only if their subgroups generated by torsion are isomorphic. Finally from this the author deduces that a countably based Abelian pro-\(p\) group with unbounded torsion contains any countably based Abelian pro-\(p\) group as a closed subgroup.