Counting subgroups and topological group topologies

Shiferaw Berhanu, W. Wistar (William) Comfort and James Dolan Reid

Vol. 116 (1985), No. 2, 217–241

DOI: 10.2140/pjm.1985.116.217

Abstract

Let G be an Abelian group with |G| = α ≥ ω, 𝒮(G) the set of subgroups of G, ℬ the set of totally bounded topological group topologies on G, ℳ(γ) the set of topological group topologies 𝒯 for which the character (= local weight) of ⟨G,𝒯⟩ is equal to γ ≥ ω, and ℬ(γ) = ℬ∩ℳ(γ). We prove algebraic results and topological results, as follows.

Algebra. Either |𝒮(G)| = 2^α or |𝒮(G)| = α. If |𝒮(G)| = α then α = ω. We describe and characterize those (countable) G such that |𝒮(G)| = ω, and we give several examples.

Topology. If γ < log(α) or γ > 2^α, then ℬ(γ) = ∅; otherwise |ℬ(γ)| = 2^α⋅γ. If γ > 2^α then ℳ(γ) = ∅; if log(α) < γ ≤ 2^α then |ℳ(γ)| = 2^αγ; and if ω ≤ γ ≤ α then |ℳ(γ)| = 2^α.

Mathematical Subject Classification 2000

Primary: 20K45

Secondary: 22A05

Milestones

Received: 17 May 1983

Published: 1 February 1985