2-closures of permutation groups

Definition2-closed actions#✓fully proved

An action of a group \(G\) on a set \(\Omega\) is \(2\)-closed when every permutation \(\sigma\) of \(\Omega\) that preserves each \(G\)-orbit on ordered pairs (that is, for all \(\alpha, \beta \in \Omega\) some \(g \in G\) satisfies \(\sigma(\alpha) = g \alpha\) and \(\sigma(\beta) = g \beta\)) is induced by an element of \(G\): there is \(g \in G\) with \(\sigma(\omega) = g \omega\) for all \(\omega \in \Omega\).

def IsTwoClosedAction (G Ω : Type) [Group G] [MulAction G Ω] : Prop :=
  ∀ σ : Equiv.Perm Ω, (∀ α β : Ω, ∃ g : G, σ α = g • α ∧ σ β = g • β) →
    ∃ g : G, ∀ ω : Ω, σ ω = g • ω

Used by 1

Used byTotally 2-closed groups

DefinitionTotally 2-closed groups#✓fully proved

A group \(G\) is totally \(2\)-closed when every faithful action of \(G\) on a set \(\Omega\) is \(2\)-closed; equivalently, in each faithful permutation representation of \(G\) on a set \(\Omega\), \(G\) is the largest subgroup of \(\mathrm{Sym}(\Omega)\) preserving each of its orbits on \(\Omega^2\).

def IsTotallyTwoClosed (G : Type) [Group G] : Prop :=
  ∀ (Ω : Type) [MulAction G Ω], FaithfulSMul G Ω → IsTwoClosedAction G Ω

Uses 1 · Used by 1

Statement uses2-closed actions

Used byKourovka Problem 20.1 (insoluble totally 2-closed groups)

DefinitionKourovka Problem 20.1 (insoluble totally 2-closed groups)#✓fully proved

The statement, asked by M. Arezoomand, M. A. Iranmanesh, C. E. Praeger and G. Tracey, that there exists a finite group \(G\) that is not soluble, is totally \(2\)-closed, and has a nontrivial nilpotent normal subgroup (equivalently, a nontrivial Fitting subgroup).

def insolubleTotallyTwoClosedWithNontrivialFitting : Prop :=
  ∃ (G : Type) (_ : Group G), Finite G ∧ ¬ Group.IsSolvable G ∧ IsTotallyTwoClosed G ∧
    ∃ N : Subgroup G, N.Normal ∧ Group.IsNilpotent N ∧ N ≠ ⊥

Uses 1 · Used by 1

Statement usesTotally 2-closed groups

Used byA finite insoluble totally 2-closed group with nontrivial Fitting subgroup

TheoremA finite insoluble totally 2-closed group with nontrivial Fitting subgroup#●ready to prove

There exists a finite group \(G\) that is not soluble, is totally \(2\)-closed, and has a nontrivial nilpotent normal subgroup (equivalently, a nontrivial Fitting subgroup).

The Kourovka Notebook (21st issue) answers Problem 20.1 "Yes, there are". Kinyon and Urban (an AI-generated solution confirmed by G. Tracey) prove that \(J_1 \times C_{13}\) is totally \(2\)-closed (Section 3.34, Lemma 3.11), insoluble and has Fitting subgroup \(C_{13}\); the Notebook also obtains \(J_1 \times C_p\) for primes \(p \in \lbrace 13, 17 \rbrace\) or \(p > 19\) from Theorem 1.1 of Arezoomand-Iranmanesh-Praeger-Tracey and Theorem 2 and Proposition 2 of Abdollahi-Arezoomand.

public theorem_wanted insolubleTotallyTwoClosedWithNontrivialFitting_holds :
    insolubleTotallyTwoClosedWithNontrivialFitting

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.1 (M. Arezoomand, M. A. Iranmanesh, C. E. Praeger, G. Tracey); resolution: M. Kinyon and J. Urban, Results on problems from the Kourovka Notebook, preprint, version 4, 19 September 2026; M. Arezoomand, M. A. Iranmanesh, C. E. Praeger and G. Tracey, Bull. Math. Sci. 14 (2024), no. 1, 2350004; A. Abdollahi and M. Arezoomand, J. Algebra Appl. 17 (2018), no. 4, 1850065, Kinyon-Urban, Section 3.34 (Problem 20.1): J1 × C13 is totally 2-closed, insoluble, with Fitting subgroup C13 (Lemma 3.11: direct products of totally 2-closed groups of coprime orders are totally 2-closed), an AI solution confirmed by G. Tracey; the Kourovka Notebook (21st issue) also derives J1 × C_p (p ∈ {13, 17} or p > 19) from Theorem 1.1 of Arezoomand-Iranmanesh-Praeger-Tracey and Theorem 2 and Proposition 2 of Abdollahi-Arezoomand, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.

Uses 1

Statement usesKourovka Problem 20.1 (insoluble totally 2-closed groups)

DefinitionThe \(2\)-closure of a permutation group#✓fully proved

For a subgroup \(G\) of the symmetric group \(\mathrm{Sym}(\Omega)\), the \(2\)-closure \(G^{(2)}\) of \(G\) is the subgroup of \(\mathrm{Sym}(\Omega)\) generated by the permutations \(\sigma\) that map every pair of points into its \(G\)-orbit, that is, such that for all \(\alpha, \beta \in \Omega\) there is \(g \in G\) with \(g(\alpha) = \sigma(\alpha)\) and \(g(\beta) = \sigma(\beta)\).

These permutations already form a subgroup: the largest subgroup of \(\mathrm{Sym}(\Omega)\) containing \(G\) with the same orbits as \(G\) on \(\Omega \times \Omega\).

def twoClosure {Ω : Type*} (G : Subgroup (Equiv.Perm Ω)) : Subgroup (Equiv.Perm Ω) :=
  Subgroup.closure {σ | ∀ a b : Ω, ∃ g ∈ G, g a = σ a ∧ g b = σ b}

Used by 1

Used byKourovka Problem 19.67

DefinitionKourovka Problem 19.67#✓fully proved

The statement that for every finite set \(\Omega\) and every solvable subgroup \(G\) of the symmetric group \(\mathrm{Sym}(\Omega)\), every composition factor of the \(2\)-closure \(G^{(2)}\) of \(G\) is cyclic or isomorphic to the alternating group \(A_n\) for some natural number \(n\).

def solvableTwoClosureFactorsCyclicOrAlternating : Prop :=
  ∀ (Ω : Type) [Finite Ω] (G : Subgroup (Equiv.Perm Ω)), Group.IsSolvable G →
    ∀ (S : Type) [Group S], IsCompositionFactor S (twoClosure G) →
      IsCyclic S ∨ ∃ n : ℕ, Nonempty (S ≃* alternatingGroup (Fin n))

Uses 2 · Used by 1

Statement usesComposition factors via subnormal subgroups · The \(2\)-closure of a permutation group

Used bySkresanov's counterexample to Kourovka Problem 19.67

TheoremSkresanov's counterexample to Kourovka Problem 19.67#●ready to prove

There exist a finite set \(\Omega\), a solvable subgroup \(G\) of the symmetric group \(\mathrm{Sym}(\Omega)\), and a composition factor \(S\) of the \(2\)-closure \(G^{(2)}\) of \(G\) such that \(S\) is not cyclic and \(S\) is not isomorphic to the alternating group \(A_n\) for any \(n\). Here \(G^{(2)}\) is the subgroup of \(\mathrm{Sym}(\Omega)\) generated by the permutations \(\sigma\) such that for all \(\alpha, \beta \in \Omega\) some \(g \in G\) has \(g(\alpha) = \sigma(\alpha)\) and \(g(\beta) = \sigma(\beta)\), and a composition factor of \(G^{(2)}\) is a simple group onto which some subnormal subgroup of \(G^{(2)}\) maps.

public theorem_wanted solvableTwoClosureFactorsCyclicOrAlternating_refuted :
    ¬ solvableTwoClosureFactorsCyclicOrAlternating

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 19.67 (I. Ponomarenko); resolution: S. V. Skresanov, Counterexamples to two conjectures in the Kourovka notebook, Algebra Logic 58 (2019), no. 3, 249-253, abstract: counterexamples to Questions 12.78 and 19.67 of the Kourovka Notebook; Kourovka Archive, Problem 19.67: No, it is not true, https://doi.org/10.1007/s10469-019-09543-1.

Uses 1

Statement usesKourovka Problem 19.67

5 fully proved2 ready to prove

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