A Topology for Primary Abelian Groups

Abstract

Let B be a p-primary abelian group without elements of infinite height, let B be a basic subgroup of G and B the torsion subgroup of the completion of B with respect to the p-adic topology. If no is the collection of all fully invariant subgroups of G with the additional property that L ∈ no implies B+L = G, for all basic subgroups B of G, then n induces a Hausdorff topology on G whose completion Ĝ is isomorphic to B (†) If H is a pure subgroup of G and if B is a basic subgroup of H, then the topology constructed for H and the relative topology agree. Homomorphisms between p-groups without elements of infinite height are continuous. Moreover, if H1 is a dense subgroup of G. and if G has no elements of infinite height, then any homomorphism of H1 onto G can be extended uniquely to a homomorphism of G onto Ĝ.