Separability of Subsets in Infinite Groups
Abstract
We study the separability of subsets of an infinite group. Given an infinite group G and subsets A,B⊂ G, we say that A and B are separated in G if there exists an infinite symmetric subset X⊂ G with e∈ X and XAX∩ B=∅. Question 17.102 of the Kourovka Notebook raises a natural cardinal question: if A and B are disjoint and |A|,|B|<|G|, must they be separated? We first give a negative answer, constructing two essentially different families of counterexamples, based respectively on rigid binary relations and on torsion-free groups whose square sets have cardinality smaller than that of the group. On this basis we introduce the W-witness set WG(A,B)={g∈ G:{g,g−1}A{g,g−1}∩ B=∅}, obtain a necessary and a sufficient condition for separability given by its size, and study the critical range. The central result of this paper is a necessary and sufficient characterization of separability: taking the inverse pairs {g,g−1} as vertices, we construct the conflict graph ΓA,B, and A and B are separated if and only if the associated graph contains an infinite independent set; separability is thereby converted into an ordinary graph-theoretic problem. From this we further derive the case of finite subsets, the case of Abelian groups, and several cardinal criteria, and we obtain, via Ramsey's theorem, a structural dichotomy for the non-separable case. In addition, we study the range of cardinalities of counterexample groups and the possible sizes of W-witness sets, and prove that no uniform necessary and sufficient criterion can depend only on |A|, |B| and structural invariants of the group, independently of the specific position of A and B.
Related
BibTeX
@misc{wan2026separability,
title = {{Separability of Subsets in Infinite Groups}},
author = {Suhui Wan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.00504},
eprint = {2608.00504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00504},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Separability of Subsets in Infinite Groups
AU - Suhui Wan
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.00504
UR - https://arxiv.org/abs/2608.00504
AB - We study the separability of subsets of an infinite group. Given an infinite group G and subsets A,B⊂ G, we say that A and B are separated in G if there exists an infinite symmetric subset X⊂ G with e∈ X and XAX∩ B=∅. Question 17.102 of the Kourovka Notebook raises a natural cardinal question: if A and B are disjoint and |A|,|B|<|G|, must they be separated? We first give a negative answer, constructing two essentially different families of counterexamples, based respectively on rigid binary relations and on torsion-free groups whose square sets have cardinality smaller than that of the group. On this basis we introduce the W-witness set W_G(A,B)={g∈ G:{g,g^−1}A{g,g^−1}∩ B=∅}, obtain a necessary and a sufficient condition for separability given by its size, and study the critical range. The central result of this paper is a necessary and sufficient characterization of separability: taking the inverse pairs {g,g^−1} as vertices, we construct the conflict graph Γ_A,B, and A and B are separated if and only if the associated graph contains an infinite independent set; separability is thereby converted into an ordinary graph-theoretic problem. From this we further derive the case of finite subsets, the case of Abelian groups, and several cardinal criteria, and we obtain, via Ramsey's theorem, a structural dichotomy for the non-separable case. In addition, we study the range of cardinalities of counterexample groups and the possible sizes of W-witness sets, and prove that no uniform necessary and sufficient criterion can depend only on |A|, |B| and structural invariants of the group, independently of the specific position of A and B.
ER -