Point stabilizers of infinite doubly transitive groups

DefinitionDoubly transitive group with infinite cyclic point stabilizer (Kourovka 10.63)#✓fully proved

The statement that there exist a group \(G\) and a faithful action of \(G\) on a set \(X\) such that for any two ordered pairs of distinct points of \(X\) some element of \(G\) maps the first to the second, and the stabilizer of some point \(x \in X\) is isomorphic to the infinite cyclic group \(\mathbb{Z}\).

def doublyTransitiveInfiniteCyclicStabilizer : Prop :=
  ∃ (G X : Type) (_ : Group G) (_ : MulAction G X), FaithfulSMul G X ∧
    MulAction.IsMultiplyPretransitive G X 2 ∧
    ∃ x : X, Nonempty (MulAction.stabilizer G x ≃* Multiplicative ℤ)
Used by 1
Used byNo doubly transitive group has an infinite cyclic point stabilizer (Kourovka 10.63)

TheoremNo doubly transitive group has an infinite cyclic point stabilizer (Kourovka 10.63)#●ready to prove

If a group \(G\) acts faithfully on a set \(X\) such that for any two ordered pairs of distinct points of \(X\) some element of \(G\) maps the first to the second, then no point stabilizer of the action is isomorphic to the infinite cyclic group \(\mathbb{Z}\).

public theorem_wanted doublyTransitiveInfiniteCyclicStabilizer_refuted :
    ¬ doublyTransitiveInfiniteCyclicStabilizer
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 10.63 (Ya. P. Sysak); resolution: V. D. Mazurov, Siberian Math. J. 31 (1990), 615-617; Gy. Károlyi, S. J. Kovács and P. P. Pálfy, Aequationes Math. 39, no. 1 (1990), 161-166, Kourovka Archive, Problem 10.63 ("No").
Uses 1
Statement usesDoubly transitive group with infinite cyclic point stabilizer (Kourovka 10.63)

DefinitionNon-periodic doubly transitive group with periodic point stabilizer (Kourovka 10.64)#✓fully proved

The statement that there exist a group \(G\) and a faithful action of \(G\) on a set \(X\) such that for any two ordered pairs of distinct points of \(X\) some element of \(G\) maps the first to the second, \(G\) has an element of infinite order, and every element of the stabilizer of some point \(x \in X\) has finite order.

def doublyTransitiveNonperiodicPeriodicStabilizer : Prop :=
  ∃ (G X : Type) (_ : Group G) (_ : MulAction G X), FaithfulSMul G X ∧
    MulAction.IsMultiplyPretransitive G X 2 ∧ (∃ g : G, ¬ IsOfFinOrder g) ∧
    ∃ x : X, ∀ h ∈ MulAction.stabilizer G x, IsOfFinOrder h
Used by 1
Used byA non-periodic doubly transitive group with a periodic point stabilizer (Kourovka 10.64)

TheoremA non-periodic doubly transitive group with a periodic point stabilizer (Kourovka 10.64)#●ready to prove

There exist a group \(G\) with an element of infinite order and a faithful action of \(G\) on a set \(X\) such that for any two ordered pairs of distinct points of \(X\) some element of \(G\) maps the first to the second, and every element of the stabilizer of some point \(x \in X\) has finite order.

public theorem_wanted doublyTransitiveNonperiodicPeriodicStabilizer_holds :
    doublyTransitiveNonperiodicPeriodicStabilizer
Sources
  • E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 10.64 (Ya. P. Sysak); resolution: M. Amelio, Non-split sharply 2-transitive groups of bounded exponent, arXiv:2509.11958 (2025), Theorem 1.3 (restated as Theorem 3.5) and Section 3.2: a non-periodic non-split sharply 2-transitive group whose involution centralizers, the point stabilizers of its action on the involutions, have bounded exponent, a positive answer to the paper's Question 2 (Kourovka 10.64); the Kourovka Notebook (21st issue) marks 10.64 solved, https://arxiv.org/abs/2509.11958.
Uses 1
Statement usesNon-periodic doubly transitive group with periodic point stabilizer (Kourovka 10.64)
2 fully proved2 ready to prove

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