[Paper Review] Subgroups of the additive group of real line

Jitender Singh|arXiv (Cornell University)|Dec 26, 2013

Holomorphic and Operator Theory2 references3 citations

TL;DR

This paper provides a new direct proof that every proper subgroup of the additive group of real numbers is either closed or dense, establishing a fundamental topological-algebraic dichotomy. It generalizes this result to topological groups, showing that any such group with this subgroup property must be either connected or totally disconnected, thereby linking subgroup topology to global group structure.

ABSTRACT

Without assuming the field structure on the additive group of real numbers $\mathbb{R}$ with the usual order $

Motivation & Objective

• To provide a direct, self-contained proof of the classical result that every proper subgroup of (R, +) is either closed or dense.
• To generalize this dichotomy to arbitrary topological groups, identifying a topological condition on subgroups that implies global connectedness or total disconnectedness.
• To clarify the structural implications of the subgroup dichotomy for the topology of the ambient group.
• To apply the result to classical number-theoretic settings, such as Kronecker's approximation theorem and the density of {sin n}.

Proposed method

• Use of topological group axioms: continuity of group operations and homogeneity via left translation.
• Application of the fact that open subgroups are closed, and that subgroups containing an open neighborhood of identity are open.
• Employment of the infimum of absolute values of nonzero elements in a subgroup to characterize closedness or denseness.
• Use of homeomorphisms (e.g., translation and scaling maps) to transfer density or closure properties across the real line.
• Proof by contradiction: assuming a non-closed, non-dense subgroup leads to a contradiction via interval covering and density of multiples.
• Generalization via Theorem 7: if all proper subgroups are closed or dense, then the group is either connected or totally disconnected.

Experimental results

Research questions

• RQ1 What topological property do proper subgroups of (R, +) necessarily satisfy, and why is this dichotomy special to R compared to R^n or R×?
• RQ2 How does the infimum of absolute values of nonzero elements in a subgroup determine whether the subgroup is closed or dense in R?
• RQ3 Can the subgroup dichotomy in R be generalized to arbitrary topological groups, and what global topological consequences follow?
• RQ4 What is the relationship between the subgroup structure and the global connectedness of a topological group?
• RQ5 How does this result apply to classical results like Kronecker’s approximation theorem?

Key findings

• Every proper subgroup H of (R, +) is either closed or dense; this is a dichotomy that holds due to the order and topology of R.
• A proper subgroup H of (R, +) is closed if and only if α = inf{|x| : x ∈ H \ {0}} ≠ 0, in which case H = αZ is cyclic.
• A proper subgroup H of (R, +) is dense if and only if α = 0, which implies the subgroup is not discrete.
• Kronecker’s approximation theorem is recovered as a corollary: for irrational α, the subgroup αZ + Z is dense in R.
• Any topological group in which every proper subgroup is either closed or dense must itself be either connected or totally disconnected.
• The real line with the lower limit topology is totally disconnected, and in this topology, every interval is totally disconnected, illustrating the failure of the subgroup dichotomy.