Hereditarily minimal topological groups

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 Zp 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 Qp ⋊ Q * p is hereditarily locally minimal, where 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.

Densely locally minimal groups

“…(2): This implication holds true for compact torsion-free groups (for a more general result see [31,Theorem B]…”

“…In [31,Proposition 3.9], we proved that a hereditarily minimal locally compact group is totally disconnected, while Example 4.6 shows that there exist densely totally minimal compact groups that are pathwise connected.…”

Topological groups all continuous automorphisms of which are open

“…Prodanov proved that an infinite compact abelian group is hereditarily minimal if and only if it is topologically isomorphic to the group Z p of p-adic integers, for some prime number p [15]. In [6], Dikranjan and Stoyanov characterized all hereditarily minimal abelian topological groups (see also [23,Fact 1.4]). It follows from this characterization that the groups of p-adic integers Z p (for a prime number p) are the only infinite locally compact, hereditarily minimal abelian groups [23,Corollary 1.5].…”

“…In [6], Dikranjan and Stoyanov characterized all hereditarily minimal abelian topological groups (see also [23,Fact 1.4]). It follows from this characterization that the groups of p-adic integers Z p (for a prime number p) are the only infinite locally compact, hereditarily minimal abelian groups [23,Corollary 1.5]. The authors of [23] made a progress in the direction of describing non-abelian hereditarily minimal topological groups.…”

“…A topological group G is called hereditarily minimal if every subgroup of G is minimal in its subspace topology [23]. Combining this definition with Theorem 3.4, we get the following…”