Separated subsets of groups

DefinitionSeparated subsets of a group#✓fully proved

Two subsets \(A\) and \(B\) of a group \(G\) are separated, in the sense of I. V. Protasov, when there is an infinite subset \(X \subseteq G\) with \(1 \in X\), \(X^{-1} = X\) and \(XAX \cap B = \emptyset\), where \(XAX = \{x a y : x, y \in X,\ a \in A\}\).

def AreSeparated {G : Type*} [Group G] (A B : Set G) : Prop :=
  ∃ X : Set G, X.Infinite ∧ (1 : G) ∈ X ∧ X⁻¹ = X ∧ Disjoint (X * A * X) B

Used by 1

Used bySeparation of small disjoint subsets of infinite groups (Kourovka 17.102)

DefinitionSeparation of small disjoint subsets of infinite groups (Kourovka 17.102)#✓fully proved

The statement, posed by I. V. Protasov, that for every infinite group \(G\) and all disjoint subsets \(A, B \subseteq G\) with \(|A| < |G|\) and \(|B| < |G|\), the sets \(A\) and \(B\) are separated.

def smallDisjointSubsetsSeparated : Prop :=
  ∀ (G : Type) [Group G] [Infinite G] (A B : Set G), Disjoint A B →
    Cardinal.mk A < Cardinal.mk G → Cardinal.mk B < Cardinal.mk G → AreSeparated A B

Uses 1 · Used by 1

Statement usesSeparated subsets of a group

Used bySmall disjoint subsets of an infinite group need not be separated

TheoremSmall disjoint subsets of an infinite group need not be separated#●ready to prove

There exist an infinite group \(G\) and disjoint subsets \(A, B \subseteq G\) with \(|A| < |G|\) and \(|B| < |G|\) such that \(A\) and \(B\) are not separated. That is, the statement of Kourovka Problem 17.102 is false.

Resolved false: Wan (Theorem 1.1) takes \(G = \mathrm{Sym}(\mathbb{N})\) and splits the set of transpositions into \(A\) and \(B\) using a graph on \(\mathbb{N}\) with trivial automorphism group, so that \(|A| = |B| = \aleph_0 < 2^{\aleph_0}\). The Kourovka Notebook (21st issue) answers 17.102 "No, not always".

public theorem_wanted smallDisjointSubsetsSeparated_refuted : ¬ smallDisjointSubsetsSeparated

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 17.102 (I. V. Protasov); resolution: S. Wan, Counterexamples to Problem 17.102 of The Kourovka Notebook, preprint, arXiv:2608.00504 (2026), Theorem 1.1 (an infinite group G with disjoint A, B, |A|, |B| < |G|, not separated; in the proof G = Sym(N) and A, B split the transpositions by a rigid graph on N); the Kourovka Notebook (21st issue) marks 17.102 solved (No, not always), https://arxiv.org/abs/2608.00504.

Uses 1

Statement usesSeparation of small disjoint subsets of infinite groups (Kourovka 17.102)

2 fully proved1 ready to prove

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