Coprime generation

DefinitionLucchini's coprime generation bound (Kourovka 16.53)#✓fully proved

The statement that for every finite group \(G\), all subgroups \(A\) and \(B\) of \(G\) of coprime orders that together generate \(G\), and every natural number \(d\), if each of \(A\) and \(B\) is generated by at most \(d\) elements, then \(G\) is generated by at most \(d + 1\) elements.

def coprimeGeneratedRankBound : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (A B : Subgroup G) (d : ℕ),
    Nat.Coprime (Nat.card A) (Nat.card B) → A ⊔ B = ⊤ →
    (∃ a : Fin d → A, Subgroup.closure (Set.range a) = ⊤) →
    (∃ b : Fin d → B, Subgroup.closure (Set.range b) = ⊤) →
    ∃ g : Fin (d + 1) → G, Subgroup.closure (Set.range g) = ⊤

Used by 1

Used byLucchini's coprime generation bound fails

TheoremLucchini's coprime generation bound fails#●ready to prove

There exist a finite group \(G\), subgroups \(A\) and \(B\) of \(G\) of coprime orders that together generate \(G\), and a natural number \(d\) such that each of \(A\) and \(B\) is generated by at most \(d\) elements, but no set of at most \(d + 1\) elements generates \(G\). That is, Kourovka Problem 16.53 has a negative answer.

Muliarchyk constructs such groups for every \(d \ge 2\), with \(A\) and \(B\) each needing exactly \(d\) generators and \(G\) needing exactly \(d + 2\); there \(A\) and \(B\) are metabelian of exponents \(6\) and \(55\) and lie inside a finite power of \(\mathrm{PSL}_2(11)\).

public theorem_wanted coprimeGeneratedRankBound_refuted : ¬ coprimeGeneratedRankBound

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 16.53 (A. Lucchini); resolution: K. Muliarchyk, Coprime generation through coset counts: a counterexample to Kourovka Notebook Problem 16.53, preprint (2026), Theorem 1 (for every d ≥ 2 finite groups A_d, B_d ≤ G_d = ⟨A_d, B_d⟩ with coprime orders, d(A_d) = d(B_d) = d and d(G_d) = d + 2); the Kourovka Notebook (21st issue) marks 16.53 solved (No, not always), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka_16_53_solution.pdf.

Uses 1

Statement usesLucchini's coprime generation bound (Kourovka 16.53)

1 fully proved1 ready to prove

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