Distinguishing sets of subdegree-finite groups

DefinitionKourovka Problem 19.89: Smith's conjecture on distinguishing sets#✓fully proved

The statement that for every countably infinite set \(\Omega\) and every subgroup \(G\) of the symmetric group \(\mathrm{Sym}(\Omega)\) that is closed in the topology of pointwise convergence and subdegree-finite (for all \(a, b \in \Omega\) the orbit of \(b\) under the stabilizer of \(a\) in \(G\) is finite), if every non-identity element of \(G\) moves infinitely many points of \(\Omega\), then there is a subset \(X \subseteq \Omega\) whose setwise stabilizer in \(G\) is trivial: the only \(g \in G\) with \(g(X) = X\) is the identity.

def closedSubdegreeFiniteDistinguishingSet : Prop :=
  ∀ (Ω : Type) [Countable Ω] [Infinite Ω] (G : Subgroup (Equiv.Perm Ω)),
    @IsClosed _ Equiv.Perm.pointwiseTopology (G : Set (Equiv.Perm Ω)) →
    (∀ a b : Ω, {c : Ω | ∃ g ∈ G, g a = a ∧ g b = c}.Finite) →
    (∀ g ∈ G, g ≠ 1 → {x : Ω | g x ≠ x}.Infinite) →
    ∃ X : Set Ω, ∀ g ∈ G, g '' X = X → g = 1

Uses 1 · Used by 1

Statement usesPointwise convergence topology on permutations

Used byLehner's theorem on distinguishing sets of subdegree-finite groups

TheoremLehner's theorem on distinguishing sets of subdegree-finite groups#●ready to prove

Let \(\Omega\) be a countably infinite set and let \(G\) be a subgroup of the symmetric group \(\mathrm{Sym}(\Omega)\) that is closed in the topology of pointwise convergence and subdegree-finite: for all \(a, b \in \Omega\) the orbit of \(b\) under the stabilizer of \(a\) in \(G\) is finite. If every non-identity element of \(G\) moves infinitely many points of \(\Omega\), then there is a subset \(X \subseteq \Omega\) whose setwise stabilizer in \(G\) is trivial: the only \(g \in G\) with \(g(X) = X\) is the identity.

Lehner proves more generally that every locally compact permutation group on a countable set in which every non-identity element moves infinitely many points has a subset with trivial setwise stabilizer; closed subdegree-finite permutation groups are locally compact.

public theorem_wanted closedSubdegreeFiniteDistinguishingSet_holds :
    closedSubdegreeFiniteDistinguishingSet

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 19.89 (S. M. Smith); resolution: F. Lehner, Asymmetric colouring of locally compact permutation groups, Bull. London Math. Soc. 55 (2023), 1874-1879 (arXiv:2208.08844), Theorem 2: a locally compact permutation group on a countable set with infinite motion (minimal degree) has an asymmetric 2-colouring, a subset with trivial setwise stabiliser; Section 2: every closed, subdegree-finite permutation group is locally compact; the Kourovka Notebook (21st issue) marks 19.89 solved: the conjecture is proved, https://arxiv.org/abs/2208.08844.

Uses 1

Statement usesKourovka Problem 19.89: Smith's conjecture on distinguishing sets

1 fully proved1 ready to prove

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