Character values of simple groups

DefinitionValues of the irreducible characters at an element#✓fully proved

For an element \(g\) of a group \(G\), the set of complex numbers \(\chi(g)\), where \(\chi\) ranges over the characters of the irreducible finite-dimensional complex representations of \(G\).

def irrCharValues {G : Type} [Group G] (g : G) : Set ℂ :=
  {z | ∃ V : FDRep ℂ G, CategoryTheory.Simple V ∧ V.character g = z}

Used by 1

Used byKourovka Problem 11.97 (simple groups with a given set of character values)

DefinitionKourovka Problem 11.97 (simple groups with a given set of character values)#✓fully proved

The statement that for every finite set \(S\) of complex numbers there is a bound \(N\) such that every finite non-abelian simple group \(G\) having an element \(g\) at which the set of values of the irreducible complex characters of \(G\) is exactly \(S\) has order at most \(N\). The problem is due to S. P. Strunkov.

"Only finitely many such groups" up to isomorphism is stated as this bound on the order.

def finitelyManySimpleGivenCharacterValues : Prop :=
  ∀ S : Set ℂ, S.Finite → ∃ N : ℕ, ∀ (G : Type) [Group G] [Finite G], IsSimpleGroup G →
    (∃ a b : G, a * b ≠ b * a) → (∃ g : G, irrCharValues g = S) → Nat.card G ≤ N

Uses 1 · Used by 1

Statement usesValues of the irreducible characters at an element

Used byKourovka Problem 11.97 has a negative answer

TheoremKourovka Problem 11.97 has a negative answer#●ready to prove

There is a finite set \(S\) of complex numbers such that for every natural number \(N\) there is a finite non-abelian simple group \(G\) with more than \(N\) elements and an element \(g \in G\) at which the set of values of the irreducible complex characters of \(G\) is exactly \(S\).

The Kourovka Archive records (V. D. Mazurov) that every complex irreducible character of \(L_2(2^m)\), \(m \ge 2\), takes a value in \(\{0, 1, -1\}\) at an involution, so some subset of \(\{0, 1, -1\}\) is the set of values at an involution for infinitely many of these non-abelian simple groups.

public theorem_wanted finitelyManySimpleGivenCharacterValues_refuted :
    ¬ finitelyManySimpleGivenCharacterValues

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 11.97 (S. P. Strunkov); resolution: V. D. Mazurov, Kourovka Archive, Problem 11.97: "No; all complex irreducible characters of the groups L_2(2^m), m ≥ 2, take the values 0, ±1 on involutions", https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka Problem 11.97 (simple groups with a given set of character values)

2 fully proved1 ready to prove

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