Locally compact groups and locally minimal group topologies

Minimal groups are Hausdorff topological groups G satisfying the open mapping theorem with respect to continuous isomorphisms, i.e., every continuous iso- morphism G → H, with H a Hausdorff topological group, is a topological isomorphism. A topological group (G,τ) is called locally minimal if there exists a neighbourhood V of the identity such that for every Hausdorff group topology σ≤τ with V ∈σ one has σ = τ. Minimal groups, as well as locally compact groups, are locally minimal. According to a well known theorem of Prodanov, every subgroup of an infinite compact abelian group K is minimal if and only if K is isomorphic to the group Zp of p-adic integers for some prime p. We find a remarkable connection of local minimality to Lie groups and p-adic numbers by means of the following results extending Prodanov’s theorem: every subgroup of a locally compact abelian group K is locally minimal if and only if either K is a Lie group, or K has an open subgroup isomorphic to Zp for some prime p. In the nonabelian case we prove that all subgroups of a connected locally compact group are locally minimal if and only if K is a Lie group, resolving Problem 7.49 from Dikranjan and Megrelishvili (2014) in the positive.