Two-generated direct powers of simple groups
DefinitionKourovka Problem 17.116 (two-generated direct powers of simple groups)#✓fully proved
The statement, in the form of its resolution, that for every non-abelian finite simple group \(G\) there is a natural number \(n\) with \(n^2 > 4|G|\), that is, \(n > 2\sqrt{|G|}\), such that the direct power \(G^n\) is generated by two elements. The problem is due to A. Erfanian and J. Wiegold.
Since \(G^n\) maps onto \(G^m\) for \(m \le n\), this says that the largest \(n(G)\) such that \(G^{n(G)}\) is 2-generated satisfies \(n(G) > 2\sqrt{|G|}\).
def directPowerTwoGeneratedBound : Prop :=
∀ (G : Type) [Group G] [Finite G], IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
∃ n : ℕ, 4 * Nat.card G < n ^ 2 ∧ ∃ x y : Fin n → G, Subgroup.closure {x, y} = ⊤Used by 1
TheoremMaróti–Tamburini bound for two-generated direct powers#●ready to prove
For every non-abelian finite simple group \(G\) there is a natural number \(n\) with \(n^2 > 4|G|\), that is, \(n > 2\sqrt{|G|}\), such that the direct power \(G^n\) is generated by two elements. This answers Kourovka Problem 17.116 positively.
Maróti and Tamburini (Theorem 1.1) prove \(2\sqrt{|G|} < h(G)\) for every non-abelian finite simple group \(G\), where \(h(G)\) is the largest \(n\) such that \(G^n\) is 2-generated; writing \(n(G)\) for this same \(h(G)\), the Kourovka Archive answers 17.116 "Yes, it is, even with \(n(G) > 2\sqrt{|G|}\)".
public theorem_wanted directPowerTwoGeneratedBound_holds : directPowerTwoGeneratedBounddef directPowerTwoGeneratedBound : Prop :=
∀ (G : Type) [Group G] [Finite G], IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
∃ n : ℕ, 4 * Nat.card G < n ^ 2 ∧ ∃ x y : Fin n → G, Subgroup.closure {x, y} = ⊤WantedExt/GroupTheory/SimpleGroupDirectPowerTwoGeneratedKourovka17Problem116Wanted.lean:43
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 17.116 (A. Erfanian, J. Wiegold); resolution: A. Maróti and M. C. Tamburini Bellani, A solution to a problem of Wiegold, Comm. Algebra 41 (2013), 34-49, Theorem 1.1 (for any non-abelian finite simple group G, 2√|G| < h(G), the largest n with G^n 2-generated); Kourovka Archive, Problem 17.116 (Yes, it is, even with n(G) > 2√|G|), https://www.renyi.hu/~maroti/centralizer12.pdf.
Uses 1
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