Locally compact groups and locally minimal group topologies

Abstract

Minimal groups are the Hausdorff topological groups G satisfying the open mapping theorem with respect to continuous isomorphisms, i.e., every continuous isomorphism G / / H, with a Hausdorff topological group H, 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 Z p 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 K is either a Lie group or K has an open subgroup isomorphic to Z p 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 in the positive Problem 7.49 from [?].

Quotients of locally minimal groups

“…A topological group G is called locally t-minimal if each Hausdorff quotient group of G is locally minimal. This property was given and used in [39] under the term local q-minimality. We prefer to use a different term (namely, local t-minimality), since one of the aims of this paper is to show that these two notions differ substantially.…”

“…The following criteria for minimality and for local minimality was established in [4,31,37] and [2] (2) When G is locally minimal and if the neighbourhood V of 1 in G witnesses both local minimality of G and local essentiality of H in G, then for every neighbourhood (a) According to [15,39,30] 3. Some general properties of the local q-, q * -and t-minimality…”

“…[15,39] If a group G has an open locally minimal subgroup, then G itself is locally minimal. In item (b) we see some examples of locally minimal groups without open locally minimal subgroups.…”