Dense subgroups and divisible quotient groups of locally compact abelian groups (Q5940623)

The author considers the relationships between the concepts of (topological) denseness and (algebraic) divisibility in locally compact abelian groups (LCA-groups). He considers the following properties in an LCA-group \(G\). Property \(D_1\): For an arbitrary subgroup \(H\), \(G/H\) is divisible implies that \(H\) is dense in \(G\). Property \(D_2\): For an arbitrary subgroup \(H\), \( H\) is dense in \(G\) implies that \(G/H\) is divisible. Property \(D_3\): An arbitrary subgroup \(H\) is dense in \(G\) if and only if \(G/H\) is divisible. -- The main results of this paper are the following.NEWLINENEWLINENEWLINETheorem 1.1. An LCA-group \(G\) has property \(D_1\) if and only if every proper closed subgroup of \(G\) is contained in a maximal subgroup.NEWLINENEWLINENEWLINETheorem 2.1. An LCA-group \(G\) has property \(D_2\) if and only if the subgroup \(pG = \{ pg \mid g \in G \}\) is open for each prime \(p\).NEWLINENEWLINENEWLINETheorem 3.1. An LCA-group \(G\) has property \(D_3\) if and only if every proper closed subgroup of \(G\) is contained in a maximal subgroup and the subgroup \(pG\) is open for each prime \(p\).NEWLINENEWLINENEWLINETheorem 3.2. A compact abelian group \(G\) has property \(D_3\) if and only if \(G \cong \prod A_p\), where \(A_p\) is a topological direct product of finitely many copies of \(J_p\) and finitely many cyclic p-groups. In particular, every compact monothetic totally disconnected group has property \(D_3\).