Successors of locally compact topological group topologies on abelian groups

Abstract

Let G be a group and σ, τ be topological group topologies on G. We say that σ is a successor of τ if σ is strictly finer than τ and there is not a group topology properly between them. In this note, we explore the existence of successor topologies in topological groups, particularly focusing on non-abelian connected locally compact groups. Our main contributions are twofold: for a connected locally compact group (G, τ ), we show that (1) if (G, τ ) is compact, then τ has a precompact successor if and only if there exists a discontinuous homomorphism from G into a simple connected compact group with dense image, and (2) if G is solvable, then τ has no successors. Our work relies on the previous characterization of locally compact group topologies on abelian groups processing successors.

M‐normal subgroups and non‐abelian lower continuous topological groups

“…Since an m‐normal subgroup of a Hausdorff topological abelian group must be a finite simple group and every complete topological group modulo a finite normal subgroup is complete, the following result generalizes [18, Lemma 2.1]. Theorem Let G be an infinite group, let σ and τ be Hausdorff topological group topologies on G such that .…”

“…Denote by * ∶ ( , ) → ( , ) the continuous extension of the identity mapping ∶ ( , ) → ( , ). Then { , } is a gap in ( ) if and only if for every factorization * = • of * , either or is a topological isomorphism, where ∶ ( , ) → and ∶ → * ( ( , )) are continuous surjective homomorphisms and is a Hausdorff topological group.Since an m-normal subgroup of a Hausdorff topological abelian group must be a finite simple group and every complete topological group modulo a finite normal subgroup is complete, the following result generalizes[18, Lemma 2.1]. Let be an infinite group, let and be Hausdorff topological group topologies on such that ≨ .…”

“…One of the main results in [18] states that a successor of a Hausdorff precompact group topology on an abelian group is still precompact. In the following example we will see that the restriction of "abelian" is necessary.…”