Thompson's problem on groups of the same type

DefinitionSame power-equation type#✓fully proved

Two groups \(G\) and \(H\) have the same power-equation type when, for every positive integer \(n\), the equation \(x^n = 1\) has the same number of solutions in \(G\) as in \(H\). An infinite set of solutions is counted as \(0\), so the notion is meant for finite groups.

def SamePowerEquationType (G H : Type*) [Group G] [Group H] : Prop :=
  ∀ n : ℕ, 0 < n →
    Nat.card {x : G // x ^ n = 1} = Nat.card {x : H // x ^ n = 1}

Used by 1

Used byCharacterization of almost simple groups by power-equation type (Kourovka 21.134(b))

DefinitionAlmost simple group#✓fully proved

A group \(G\) is almost simple when there is a nonabelian finite simple group \(S\) such that \(G\) is isomorphic to a subgroup \(A\) of the automorphism group \(\mathrm{Aut}(S)\) that contains the inner automorphism group \(\mathrm{Inn}(S)\).

def IsAlmostSimple (G : Type u) [Group G] : Prop :=
  ∃ (S : Type u) (groupStructure : Group S) (finiteS : Finite S)
      (simpleS : IsSimpleGroup S),
    letI := groupStructure
    letI := finiteS
    letI := simpleS
    ¬IsMulCommutative S ∧
      ∃ A : Subgroup (MulAut S),
        MonoidHom.range (MulAut.conj : S →* MulAut S) ≤ A ∧
          Nonempty (G ≃* A)

Used by 1

Used byCharacterization of almost simple groups by power-equation type (Kourovka 21.134(b))

DefinitionCharacterization of almost simple groups by power-equation type (Kourovka 21.134(b))#✓fully proved

The conjecture that for all finite groups \(G\) and \(H\), if \(G\) is almost simple and \(G\) and \(H\) have the same power-equation type, then \(G\) is isomorphic to \(H\).

def conjecture : Prop :=
  ∀ (G H : Type*) [Group G] [Finite G] [Group H] [Finite H],
    IsAlmostSimple G → SamePowerEquationType G H → Nonempty (G ≃* H)

Uses 2 · Used by 1

Statement usesAlmost simple group · Same power-equation type

Used byPower-equation type does not characterize almost simple groups

TheoremPower-equation type does not characterize almost simple groups#●ready to prove

There exist finite groups \(G\) and \(H\) such that \(G\) is almost simple, \(G\) and \(H\) have the same power-equation type, and \(G\) is not isomorphic to \(H\). That is, the conjecture of Kourovka Problem 21.134(b) is false.

Kourovka Notebook v46 records that J. G. Thompson had already given a negative answer to part (b).

theorem_wanted conjecture_refuted : ¬ conjecture

Sources

• Yu Li and Wujie Shi, A Note on Thompson Problem, arXiv:2303.09460v1 (2023); published in Ricerche di Matematica 74(1) (2025), 559-563, https://arxiv.org/abs/2303.09460v1; The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, Problem 21.134, arXiv:1401.0300v46, https://arxiv.org/abs/1401.0300v46.

Uses 1

Statement usesCharacterization of almost simple groups by power-equation type (Kourovka 21.134(b))

DefinitionGroups of the same type#✓fully proved

Two groups \(G\) and \(H\) have the same type when, for every integer \(n \ge 1\), the equation \(x^n = 1\) has the same number of solutions in \(G\) as in \(H\), where an infinite set of solutions counts as having \(0\) elements.

def SameType (G H : Type*) [Group G] [Group H] : Prop :=
  ∀ n : ℕ, 0 < n → Nat.card {x : G // x ^ n = 1} = Nat.card {x : H // x ^ n = 1}

Used by 1

Used byKourovka Problem 12.37 (J. G. Thompson)

DefinitionKourovka Problem 12.37 (J. G. Thompson)#✓fully proved

Every finite group of the same type as a finite soluble group is soluble: if \(G\) and \(H\) are finite groups of the same type and \(H\) is soluble, then \(G\) is soluble.

def solubleOfSameTypeAsSoluble : Prop :=
  ∀ (G H : Type) [Group G] [Finite G] [Group H] [Finite H],
    Group.IsSolvable H → SameType G H → Group.IsSolvable G

Uses 1 · Used by 1

Statement usesGroups of the same type

Used bySolubility is not determined by the type

TheoremSolubility is not determined by the type#●ready to prove

The answer to Kourovka Problem 12.37 is negative: there are finite groups \(G\) and \(H\) of the same type such that \(H\) is soluble and \(G\) is not soluble.

public theorem_wanted solubleOfSameTypeAsSoluble_refuted : ¬ solubleOfSameTypeAsSoluble

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 12.37 (J. G. Thompson; A. S. Kondratiev, W. J. Shi); resolution: P. Piwek, Algebra Number Theory 19 (2025), no. 8, 1663-1670, the Kourovka Notebook (21st issue) marks 12.37 solved: No, not always.

Uses 1

Statement usesKourovka Problem 12.37 (J. G. Thompson)

5 fully proved2 ready to prove

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