Sharply 2-transitive groups

DefinitionSharply doubly transitive action#✓fully proved

An action of a group \(G\) on a set \(\Omega\) is sharply doubly transitive when for all points \(\alpha, \beta, \gamma, \delta \in \Omega\) with \(\alpha \ne \beta\) and \(\gamma \ne \delta\) there is exactly one \(g \in G\) with \(g \cdot \alpha = \gamma\) and \(g \cdot \beta = \delta\).

def IsSharplyTwoTransitive (G Ω : Type*) [Group G] [MulAction G Ω] : Prop :=
  ∀ α β γ δ : Ω, α ≠ β → γ ≠ δ → ∃! g : G, g • α = γ ∧ g • β = δ

Used by 1

Used byMazurov's question on sharply 2-transitive groups (Kourovka 11.52)

DefinitionMazurov's question on sharply 2-transitive groups (Kourovka 11.52)#✓fully proved

The statement that every group \(G\) acting sharply doubly transitively on a set \(\Omega\) with at least two points has a nontrivial abelian normal subgroup.

def sharplyTwoTransitiveHasAbelianNormal : Prop :=
  ∀ (G Ω : Type) [Group G] [MulAction G Ω], Nontrivial Ω → IsSharplyTwoTransitive G Ω →
    ∃ N : Subgroup G, N.Normal ∧ N ≠ ⊥ ∧ ∀ x ∈ N, ∀ y ∈ N, x * y = y * x

Uses 1 · Used by 1

Statement usesSharply doubly transitive action

Used byA sharply 2-transitive group without nontrivial abelian normal subgroups

TheoremA sharply 2-transitive group without nontrivial abelian normal subgroups#●ready to prove

The statement of Kourovka Problem 11.52 is false: there is a group \(G\) acting sharply doubly transitively on a set \(\Omega\) with at least two points such that no nontrivial normal subgroup of \(G\) is abelian.

Rips, Segev and Tent construct such a group; the Kourovka Archive answers 11.52 "No, not every". For finite groups the answer is yes, by Frobenius's theorem.

public theorem_wanted sharplyTwoTransitiveHasAbelianNormal_refuted :
    ¬ sharplyTwoTransitiveHasAbelianNormal

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 11.52 (V. D. Mazurov); resolution: E. Rips, Y. Segev and K. Tent, J. Eur. Math. Soc. 19, no. 10 (2017), 2895-2910, Kourovka Archive, Problem 11.52: "No, not every", https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesMazurov's question on sharply 2-transitive groups (Kourovka 11.52)

2 fully proved1 ready to prove

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