Conjugacy class sizes of p-groups

DefinitionPrescribed sets of conjugacy class sizes of finite p-groups (Kourovka 14.77)#✓fully proved

The statement that for every prime \(p\) and every finite set \(X\) of natural numbers that contains \(1\) and consists of powers of \(p\), there is a finite \(p\)-group \(G\) whose set of conjugacy class sizes \(\{|C| : C \text{ a conjugacy class of } G\}\) equals \(X\).

def everyPowerSetIsClassSizeSet : Prop :=
  ∀ p : ℕ, p.Prime → ∀ X : Finset ℕ, 1 ∈ X → (∀ x ∈ X, ∃ k : ℕ, x = p ^ k) →
    ∃ (G : Type) (_ : Group G) (_ : Finite G), IsPGroup p G ∧
      Set.range (fun c : ConjClasses G => Nat.card c.carrier) = (X : Set ℕ)

Used by 1

Used byCossey–Hawkes theorem on conjugacy class sizes of p-groups

TheoremCossey–Hawkes theorem on conjugacy class sizes of p-groups#●ready to prove

For every prime \(p\) and every finite set \(X\) of natural numbers that contains \(1\) and consists of powers of \(p\), there is a finite \(p\)-group whose set of conjugacy class sizes is exactly \(X\). This answers Kourovka Problem 14.77 (Sangroniz) in the affirmative.

public theorem_wanted everyPowerSetIsClassSizeSet_holds : everyPowerSetIsClassSizeSet

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 14.77 (J. Sangroniz); resolution: J. Cossey and T. Hawkes, Proc. Amer. Math. Soc. 128 (2000), no. 1, 49-51, Kourovka Archive, Problem 14.77 (answer: Yes, it is).

Uses 1

Statement usesPrescribed sets of conjugacy class sizes of finite p-groups (Kourovka 14.77)

DefinitionNumber of conjugacy classes of maximum size#✓fully proved

For a group \(G\), the number of conjugacy classes \(c\) of \(G\) such that every conjugacy class of \(G\) has size at most the size of \(c\). It is meant for finite groups: sizes and counts are taken in \(\mathbb{N}\), where an infinite set has size \(0\).

noncomputable def numMaxSizeConjClasses (G : Type*) [Group G] : ℕ :=
  Nat.card {c : ConjClasses G // ∀ d : ConjClasses G, Nat.card d.carrier ≤ Nat.card c.carrier}

Used by 1

Used byOdd \(p\)-groups with \(p - 1\) conjugacy classes of maximum size (Kourovka 14.92)

DefinitionOdd \(p\)-groups with \(p - 1\) conjugacy classes of maximum size (Kourovka 14.92)#✓fully proved

The statement that there exist an odd prime \(p\) and a finite \(p\)-group with exactly \(p - 1\) conjugacy classes of maximum size.

def oddPGroupWithPredMaxSizeConjClasses : Prop :=
  ∃ p : ℕ, p.Prime ∧ Odd p ∧ ∃ (G : Type) (_ : Group G) (_ : Finite G), IsPGroup p G ∧
    numMaxSizeConjClasses G = p - 1

Uses 1 · Used by 1

Statement usesNumber of conjugacy classes of maximum size

Used bySome odd \(p\)-group has exactly \(p - 1\) conjugacy classes of maximum size

TheoremSome odd \(p\)-group has exactly \(p - 1\) conjugacy classes of maximum size#●ready to prove

There exist an odd prime \(p\) and a finite \(p\)-group with exactly \(p - 1\) conjugacy classes of maximum size.

public theorem_wanted oddPGroupWithPredMaxSizeConjClasses_holds :
    oddPGroupWithPredMaxSizeConjClasses

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 14.92 (G. Fernández-Alcober); resolution: A. Jaikin-Zapirain, M. F. Newman and E. A. O'Brien, p-groups having the minimal number of conjugacy classes of maximal size, Israel J. Math. 172 (2009), no. 1, 119-123, abstract (a 3-group with exactly p - 1 = 2 conjugacy classes of the largest size); Kourovka Archive, Problem 14.92 (answer: Yes, there are, for p = 3), http://verso.mat.uam.es/~andrei.jaikin/preprints/articulos.htm.

Uses 1

Statement usesOdd \(p\)-groups with \(p - 1\) conjugacy classes of maximum size (Kourovka 14.92)

3 fully proved2 ready to prove

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