Thompson's conjecture on conjugacy class sizes

DefinitionThe set of conjugacy class sizes#✓fully proved

For a group \(G\), \(N(G)\) is the set of natural numbers \(|\{y \in G : y \text{ is conjugate to } x\}|\) for \(x \in G\), that is, the set of sizes of the conjugacy classes of \(G\), where an infinite class counts as size \(0\).

def classSizes (G : Type*) [Group G] : Set ℕ :=
  Set.range fun x : G => Nat.card {y : G // IsConj x y}

Used by 1

Used byThompson's conjecture on conjugacy class sizes

DefinitionThompson's conjecture on conjugacy class sizes#✓fully proved

Write \(N(G)\) for the set of sizes of the conjugacy classes of a group \(G\). Thompson's conjecture is the statement that for all finite groups \(G\) and \(H\), if \(G\) is simple and non-abelian, \(H\) has trivial centre, and \(N(G) = N(H)\), then \(G\) and \(H\) are isomorphic.

This is Kourovka Notebook Problem 12.38 (J. G. Thompson).

def simpleGroupRecognizedByClassSizes : Prop :=
  ∀ (G H : Type) [Group G] [Finite G] [Group H] [Finite H], IsSimpleGroup G →
    (∃ a b : G, a * b ≠ b * a) → Subgroup.center H = ⊥ → classSizes G = classSizes H →
    Nonempty (G ≃* H)

Uses 1 · Used by 1

Statement usesThe set of conjugacy class sizes

Used byThompson's conjecture on conjugacy class sizes is true

TheoremThompson's conjecture on conjugacy class sizes is true#●ready to prove

For all finite groups \(G\) and \(H\), if \(G\) is simple and non-abelian, \(H\) has trivial centre, and \(G\) and \(H\) have the same set of conjugacy class sizes, then \(G\) and \(H\) are isomorphic.

Resolved true: the Kourovka Archive answers 12.38 "Yes, it is true" (Thompson's conjecture); the final step is I. B. Gorshkov (Commun. Algebra 47 (2019)), completing a case-by-case proof over the finite simple groups by many authors.

public theorem_wanted simpleGroupRecognizedByClassSizes_holds : simpleGroupRecognizedByClassSizes

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 12.38 (J. G. Thompson; A. S. Kondratiev, W. J. Shi); resolution: I. B. Gorshkov, Commun. Algebra 47 (2019), no. 12, 5192-5206 (final step), Kourovka Archive, Problem 12.38: Yes, it is true; earlier steps by many authors cited there.

Uses 1

Statement usesThompson's conjecture on conjugacy class sizes

2 fully proved1 ready to prove

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