Densely locally minimal groups

Abstract

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups Zp of p-adic integers. In [31], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups.We prove that in case that a topological abelian group G is either compact or connected locally compact, then G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Zp for some prime p. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to Zp. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to Zp.

C-Minimal topological groups

“…Example 3.4. By [33,Example 4.6], the special orthogonal group G = SO(n, R), where n ≥ 5, is densely totally minimal but not hereditarily minimal. Being compact, G is also c-totally minimal.…”

“…Dikranjan, Toller and the authors introduced the following two concepts in [32,Definition 1.2] and [33,Definition 1.5], respectively. Definition 1.3.…”

Quotients of locally minimal groups

“…Locally compact groups and normed vector spaces are locally minimal [30]. Further details on locally minimal groups can be found in [15,1,2,40,41]. The relevant permanence properties of local minimality related to the passage to closed or dense subgroups were largely studied in these papers.…”