Generation by elements of prime-power order

DefinitionKourovka Problem 18.52 (generation by two elements of prime-power order)#✓fully proved

The statement that every finite simple group \(G\) is generated by two elements \(x, y \in G\) whose orders are prime powers, that is, each of the form \(p^k\) with \(p\) prime and \(k \ge 1\) (the primes and exponents may differ between \(x\) and \(y\) and depend on the group). The problem is due to J. Krempa.

def simpleGroupGeneratedByPrimePowerOrderElements : Prop :=
  ∀ (G : Type) [Group G] [Finite G], IsSimpleGroup G →
    ∃ x y : G, IsPrimePow (orderOf x) ∧ IsPrimePow (orderOf y) ∧ Subgroup.closure {x, y} = ⊤

Used by 1

Used byFinite simple groups are generated by two elements of prime-power order

TheoremFinite simple groups are generated by two elements of prime-power order#●ready to prove

Every finite simple group \(G\) is generated by two elements \(x, y \in G\) whose orders are prime powers, that is, each of the form \(p^k\) with \(p\) prime and \(k \ge 1\), where the prime and the exponent may differ between \(x\) and \(y\). This answers Kourovka Problem 18.52 positively.

The Kourovka Archive answers 18.52 "Yes, it is" (mod CFSG): by C. S. H. King (2017) every finite (non-abelian) simple group is generated by an involution and an element of prime order; a cyclic group of prime order is generated by one element of prime order.

public theorem_wanted simpleGroupGeneratedByPrimePowerOrderElements_holds :
    simpleGroupGeneratedByPrimePowerOrderElements

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 18.52 (J. Krempa); resolution: C. S. H. King, J. Algebra 478 (2017), 153-173, Kourovka Archive, Problem 18.52: Yes, it is (mod CFSG); moreover every finite simple group is generated by an involution and an element of prime order, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka Problem 18.52 (generation by two elements of prime-power order)

1 fully proved1 ready to prove

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