Largest conjugacy class of perfect groups

DefinitionKourovka 20.30: the order of a perfect centreless group and its largest conjugacy class#✓fully proved

The proposition that every finite group \(G\) that is perfect (\([G, G] = G\)) and has trivial centre contains an element \(x\) with \(|G| \le |x^G|^2\), where \(x^G\) is the conjugacy class of \(x\). Equivalently, \(|G| \le n^2\), where \(n\) is the largest size of a conjugacy class of \(G\).

def perfectCentrelessOrderLeMaxClassSq : Prop :=
  ∀ (G : Type) [Group G] [Finite G], Group.IsPerfect G → Subgroup.center G = ⊥ →
    ∃ x : G, Nat.card G ≤ (conjugatesOf x).ncard ^ 2

Used by 1

Used byPerfect centreless groups need not have order at most the square of the largest class size

TheoremPerfect centreless groups need not have order at most the square of the largest class size#●ready to prove

It is not true that every finite group \(G\) that is perfect (\([G, G] = G\)) and has trivial centre contains an element \(x\) with \(|G| \le |x^G|^2\), where \(x^G\) is the conjugacy class of \(x\). That is, some perfect centreless finite group \(G\) satisfies \(|G| > n^2\), where \(n\) is the largest size of a conjugacy class of \(G\).

Muliarchyk (Theorem 1.1) constructs a perfect centreless finite group of order \(120 \cdot 5^{68}\) whose largest conjugacy class has size \(30 \cdot 5^{32}\), so \(|G| / n^2 = 250/3 > 1\); the 21st issue of the Kourovka Notebook marks 20.30 solved ("No, not always"). The preprint acknowledges assistance from ChatGPT.

public theorem_wanted perfectCentrelessOrderLeMaxClassSq_refuted :
    ¬ perfectCentrelessOrderLeMaxClassSq

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.30 (P. M. Neumann and M. R. Vaughan-Lee; I. B. Gorshkov); resolution: K. Muliarchyk, Perfect centreless groups with large centralizers, preprint (2026), Kourovka Notebook site, Theorem 1.1: an explicit perfect centreless finite group of order 120·5^68 whose largest conjugacy class has size 30·5^32, so |G|/n² = 250/3 > 1 (built from a class-three Baker-Campbell-Hausdorff group over F₅ and an SL₂(F₅)-action); Theorem 6.1: no bound |G| ≤ C n^c holds for all perfect centreless finite groups; Kourovka Notebook (21st issue), Problem 20.30: No, not always, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka_20_30_solution.pdf.

Uses 1

Statement usesKourovka 20.30: the order of a perfect centreless group and its largest conjugacy class

1 fully proved1 ready to prove

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