Hereditarily minimal topological groups

Abstract

Abstract We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal. In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups ℤ p {\mathbb{Z}_{p}} of p-adic integers. We extend Prodanov’s theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups, we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that, in particular, they are always compact and metabelian. The proofs involve the (hereditarily) locally minimal groups, introduced similarly. In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group ℚ p ⋊ ℚ p * {\mathbb{Q}_{p}\rtimes\mathbb{Q}_{p}^{*}} is hereditarily locally minimal, where ℚ p * {\mathbb{Q}_{p}^{*}} is the multiplicative group of non-zero p-adic numbers acting on the first component by multiplication. Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.