Class number of a finite group

DefinitionHeffernan–MacHale problem on groups of maximal order with a given class number#✓fully proved

The proposition that for every integer \(k \ge 1\) such that some insoluble finite group has exactly \(k\) conjugacy classes, every finite group \(G\) with exactly \(k\) conjugacy classes whose order is at least the order of every finite group with exactly \(k\) conjugacy classes is insoluble. This is Kourovka Problem 17.77 (R. Heffernan, D. MacHale).

def maximalOrderClassNumberInsoluble : Prop :=
  ∀ k : ℕ, 0 < k →
    (∃ (H : Type) (_ : Group H) (_ : Finite H),
      ¬ Group.IsSolvable H ∧ Nat.card (ConjClasses H) = k) →
    ∀ (G : Type) [Group G] [Finite G], Nat.card (ConjClasses G) = k →
      (∀ (H : Type) [Group H] [Finite H], Nat.card (ConjClasses H) = k → Nat.card H ≤ Nat.card G) →
        ¬ Group.IsSolvable G

Used by 1

Used byA finite group of maximal order with a given class number can be soluble

TheoremA finite group of maximal order with a given class number can be soluble#●ready to prove

There are an integer \(k \ge 1\), an insoluble finite group with exactly \(k\) conjugacy classes, and a soluble finite group \(G\) with exactly \(k\) conjugacy classes whose order is at least the order of every finite group with exactly \(k\) conjugacy classes. So it is not true that whenever some insoluble finite group has exactly \(k \ge 1\) conjugacy classes, every finite group of maximal order with exactly \(k\) conjugacy classes is insoluble.

public theorem_wanted maximalOrderClassNumberInsoluble_refuted : ¬ maximalOrderClassNumberInsoluble

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 17.77 (R. Heffernan, D. MacHale); resolution: A. Vera-López and J. Sangroniz, Math. Nachr. 280 (2007), 676-694, Kourovka Archive, Problem 17.77 (No, it is not; see Table 1 in Vera-López-Sangroniz).

Uses 1

Statement usesHeffernan–MacHale problem on groups of maximal order with a given class number

DefinitionLiebeck–Pyber problem on the class number and nilpotent subgroups#✓fully proved

The proposition that every finite group \(G\) has a nilpotent subgroup \(N\) whose order \(|N|\) is at least the number \(k(G)\) of conjugacy classes of \(G\). This is Kourovka Problem 14.54 (M. W. Liebeck, L. Pyber).

def classNumberAtMostNilpotentSubgroupOrder : Prop :=
  ∀ (G : Type) [Group G] [Finite G], ∃ N : Subgroup G, Group.IsNilpotent N ∧
    Nat.card (ConjClasses G) ≤ Nat.card N

Used by 1

Used byThe class number can exceed the order of every nilpotent subgroup

TheoremThe class number can exceed the order of every nilpotent subgroup#●ready to prove

There is a finite group \(G\) such that the number \(k(G)\) of conjugacy classes of \(G\) is greater than \(|N|\) for every nilpotent subgroup \(N\) of \(G\).

public theorem_wanted classNumberAtMostNilpotentSubgroupOrder_refuted :
    ¬ classNumberAtMostNilpotentSubgroupOrder

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.54 (M. W. Liebeck, L. Pyber); resolution: S. Tertooy, Class numbers and nilpotent subgroups of PGL(2,q), preprint, arXiv:2606.21404 (2026), abstract: for certain odd prime powers q, k(PGL(2,q)) exceeds the order of every nilpotent subgroup; the Kourovka Notebook (21st issue) marks 14.54 solved (No, not always), https://arxiv.org/abs/2606.21404.

Uses 1

Statement usesLiebeck–Pyber problem on the class number and nilpotent subgroups

2 fully proved2 ready to prove

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