Involutions in periodic groups

DefinitionKourovka Problem 10.62 on involution classes with products of odd order#✓fully proved

The proposition (Kourovka 10.62, A. I. Sozutov) that there is a periodic group \(G\) that has no subgroup of index \(2\) and contains an involution \(x\) such that the conjugacy class \(X\) of \(x\) generates \(G\) and the product \(yz\) has odd order for all \(y, z \in X\).

The condition includes \(y = z\), where \(yz = 1\) has order \(1\).

def periodicGroupGeneratedByOddInvolutionClass : Prop :=
  ∃ (G : Type) (_ : Group G), (∀ g : G, IsOfFinOrder g) ∧ (∀ H : Subgroup G, H.index ≠ 2) ∧
    ∃ x : G, orderOf x = 2 ∧ Subgroup.closure {y | IsConj x y} = ⊤ ∧
      ∀ y z : G, IsConj x y → IsConj x z → Odd (orderOf (y * z))

Used by 1

Used byPositive answer to Kourovka Problem 10.62

TheoremPositive answer to Kourovka Problem 10.62#●ready to prove

There is a periodic group \(G\) that has no subgroup of index \(2\) and contains an involution \(x\) such that the conjugacy class \(X\) of \(x\) generates \(G\) and the product \(yz\) has odd order for all \(y, z \in X\).

Kinyon and Urban's 2026 preprint (an AI-generated solution confirmed by M. Amelio; Section 3.7, Proposition 3.5) constructs, for every sufficiently large odd prime \(p\), an infinite three-generated perfect group of exponent \(2p\) whose involutions form one conjugacy class generating the group, the product of any two distinct involutions having order \(p\); being perfect, it has no subgroup of index \(2\). The Kourovka Notebook (21st issue) marks 10.62 solved, "Example constructed".

public theorem_wanted periodicGroupGeneratedByOddInvolutionClass_holds :
    periodicGroupGeneratedByOddInvolutionClass

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 10.62 (A. I. Sozutov); resolution: M. Kinyon and J. Urban (with Codex gpt-6-astra), Results on problems from the Kourovka Notebook, preprint (2026), Section 3.7, Proposition 3.5 (built on Amelio's geometric construction); AI solution confirmed by M. Amelio; the Kourovka Notebook (21st issue) marks 10.62 solved, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.

Uses 1

Statement usesKourovka Problem 10.62 on involution classes with products of odd order

1 fully proved1 ready to prove

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