Products of conjugacy classes in alternating groups

DefinitionProducts of three large conjugacy classes in alternating groups (Kourovka 20.23)#✓fully proved

The statement that there are a real number \(\delta\) with \(0 < \delta < 1\) and a natural number \(N\) such that for every \(n \ge N\) and all conjugacy classes \(A\), \(B\), \(C\) of the alternating group \(\mathrm{Alt}(n)\), each of size at least \(|\mathrm{Alt}(n)|^{\delta}\), the product set \(ABC\) is all of \(\mathrm{Alt}(n)\).

def productOfThreeLargeClassesIsAlternatingGroup : Prop :=
  ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∃ N : ℕ, ∀ n : ℕ, N ≤ n →
    ∀ A B C : ConjClasses (alternatingGroup (Fin n)),
      (Nat.card (alternatingGroup (Fin n)) : ℝ) ^ δ ≤ Nat.card A.carrier →
      (Nat.card (alternatingGroup (Fin n)) : ℝ) ^ δ ≤ Nat.card B.carrier →
      (Nat.card (alternatingGroup (Fin n)) : ℝ) ^ δ ≤ Nat.card C.carrier →
      A.carrier * B.carrier * C.carrier = Set.univ

Used by 1

Used byThree large conjugacy classes multiply to the whole alternating group

TheoremThree large conjugacy classes multiply to the whole alternating group#●ready to prove

There are a real number \(\delta\) with \(0 < \delta < 1\) and a natural number \(N\) such that for every \(n \ge N\) and all conjugacy classes \(A\), \(B\), \(C\) of the alternating group \(\mathrm{Alt}(n)\), each of size at least \(|\mathrm{Alt}(n)|^{\delta}\), the product set \(ABC\) is all of \(\mathrm{Alt}(n)\). This is the positive answer to Kourovka Problem 20.23.

Dona (2025, Theorem 1.2) proves this with the size bound written as \(|\mathrm{Alt}(n)|^{1-\varepsilon}\) for a small \(\varepsilon > 0\).

public theorem_wanted productOfThreeLargeClassesIsAlternatingGroup_holds :
    productOfThreeLargeClassesIsAlternatingGroup

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.23 (M. Garonzi, A. Maróti); resolution: D. Dona, Products of three conjugacy classes in the alternating group, preprint, arXiv:2505.06012 (2025), Theorem 1.2: there are δ > 0 and n₀ such that for n ≥ n₀ any three conjugacy classes of G = Alt(n) of size at least |G|^(1-δ) satisfy C₁C₂C₃ = G (a positive answer to Problem 20.23 with exponent 1 - δ); the Kourovka Notebook (21st issue) marks 20.23 solved, https://arxiv.org/abs/2505.06012.

Uses 1

Statement usesProducts of three large conjugacy classes in alternating groups (Kourovka 20.23)

1 fully proved1 ready to prove

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