Topological groups with a countable set of monothetic subgroups (Q1589169)

Let \(G\) be a locally compact group and \(L(G)\) the lattice of its closed subgroups. By \(\lambda (G)\) is denoted the cardinal of \(L(G);\) by \(\mu (G)\) the cardinal of the set of monothetic subgroups (i.e. closed subgroups topologically generated by one element); by \(\beta (G)\) the cardinal of closed subgroups topologically isomorphic to \(\mathbb{Z}_{p}\) or \(\mathbb{Z} /(p^{n})\) for \(n\in \mathbb{N}^{+}\) (\(p\) is prime and \(\mathbb{Z}_{p}\) is the group of \(p\)-adic integers). It is proved (Theorem 1) that if \(G\) is an inductive compact \(p\)-group, then \( \beta (G)\leq \aleph _{0}\) if and only if the subset \(\Omega (G)\) of all periodic elements is a countable discrete subgroup and the quotient group \( G/\Omega (G)\) is topologically isomorphic to the additive group \(\mathbb{Q}_{p}\) of \(p\)-adic numbers (\(p\) is prime), \(\mathbb{Z}_{p}\) or \(\{e\}.\) Also a characterization is given of inductive pronilpotent locally compact groups \(G \) for which \(\beta (G)\leq \aleph _{0}\) or if the cardinal of all closed Abelian subgroups is \(\leq \aleph _{0}.\)