Counterexamples to Problem 17.102 of the Kourovka Notebook: Negation and Discussion of Separability Conditions in Infinite Groups

Abstract

This paper gives a negative answer to Problem 17.102 of The Kourovka Notebook: there exist an infinite group $G$ and disjoint subsets $A, B \subset G$ satisfying $|A|, |B| < |G|$, yet $A$ and $B$ are not separable in $G$. We introduce the witness set $W_G(A, B)$ and prove a necessary condition and a sufficient condition for separability. Using the torsion-free group constructed by Newelski and rigid binary relations, we obtain two kinds of counterexamples in which $W_G(A, B)$ is finite. Furthermore, we construct examples in which $W_G(A, B)$ is infinite yet separation remains impossible, showing that the sufficient condition cannot be weakened.