Locally compact abelian $p$-groups

A locally compact abelian group is called periodic if it is totally disconnected and is a directed union of its compact subgroups. Various aspects of abelian periodic groups are considered such as – decomposing them into local products of their Sylow $p$- subgroups, – providing new descriptions of periodic abelian torsion groups, and of – periodic abelian divisible groups and their torsion-free and their torsion components, – reviewing splitting theorems, notably for finite rank pure subgroups of (almost) finite exponent $p$-groups, – providing a definition of a general $p$-rank for all locally compact abelian $p$-groups.