Locally compact abelian \(p\)-groups (Q2415928)

In this interesting and well-written paper the authors study various aspects of \textit{ periodic} locally compact abelian (lca) groups. An lca group \(G\) is called \textit{ periodic} if it is totally disconnected and is a direct union of its compact subgroups. It is proved that in every lca \(p\)-group having approximately finite exponent, each finitely generated subgroup is contained in a finitely generated (algebraic and topological) direct summand of the same rank. It is shown that any lca \(p\)-group is an open subgroup of its divisible hull which is a torsion-free divisible \(p\)-group. The character group of an lca torsion-free divisible \(p\)-group \(G\) is divisible if and only if \(G\cong \mathbb{Q}_p^n\) for some nonnegative integer \(n\). Any torsion-free periodic \(G\) is isomorphic to \(\mathbb{Q}\otimes \prod_{p\in \mathbb{P}}\mathbb{Z}_p^{I_p}\), where \(\mathbb{P}\) is the family of prime numbers and \(\{ I_p: p\in \mathbb{P}\}\) is a family of sets. Every closed divisible subgroup of a torsion-free lca \(p\)-group is a direct summand (algebraically and topologically). If there is a compact open subgroup \(C\) of a torsion-free lca \(p\)-group \(G\) such that the discrete factor group \(G/C\) is divisible, then there is a profinite subgroup \(R\) of \(G\) such that \(G=R\oplus div(G)\) algebraically and topologically. Many other structural theorems are obtained. The authors provide new descriptions of periodic abelian torsion groups and a definition of a general \(p\)-rank for all lca \(p\)-groups.