Products of maximal subgroups

DefinitionNon-soluble products of non-conjugate maximal subgroups (Kourovka 10.34, Monakhov)#✓fully proved

The statement that there is a finite group \(G\) that is not soluble such that \(G = MN\) for every two maximal subgroups \(M\) and \(N\) of \(G\) that are not conjugate in \(G\). Here \(MN\) is the set of products \(mn\) with \(m \in M\) and \(n \in N\).

def nonsolubleProductOfNonconjugateMaximals : Prop :=
  ∃ (G : Type) (_ : Group G) (_ : Finite G), ¬ Group.IsSolvable G ∧
    ∀ M N : Subgroup G, IsCoatom M → IsCoatom N →
      (∀ g : G, N ≠ M.map (MulAut.conj g).toMonoidHom) → (M : Set G) * (N : Set G) = Set.univ

Used by 1

Used byFinite groups that are the product of every two non-conjugate maximal subgroups are soluble

TheoremFinite groups that are the product of every two non-conjugate maximal subgroups are soluble#●ready to prove

There is no finite group \(G\) that is not soluble such that \(G = MN\) for every two maximal subgroups \(M\) and \(N\) of \(G\) that are not conjugate in \(G\); here \(MN\) is the set of products \(mn\) with \(m \in M\) and \(n \in N\). This answers Kourovka Problem 10.34 in the negative.

Sater (arXiv:2608.02970) and Li and Yang (arXiv:2608.19478) prove that a finite group with this property is soluble (the almost simple case is due to Tikhonenko and Tyutyanov, 2010). The Kourovka Notebook (21st issue) marks 10.34 solved, "No, there are no such groups".

public theorem_wanted nonsolubleProductOfNonconjugateMaximals_refuted :
    ¬ nonsolubleProductOfNonconjugateMaximals

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 10.34 (V. S. Monakhov); resolution: R. Sater, Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble, arXiv:2608.02970 (2026); J. Li and Y. Yang, Products of nonconjugate maximal subgroups, arXiv:2608.19478 (2026), main theorems: a finite group with MN = G for all non-conjugate maximal subgroups M, N is soluble; the Kourovka Notebook (21st issue) marks 10.34 solved, https://arxiv.org/abs/2608.19478.

Uses 1

Statement usesNon-soluble products of non-conjugate maximal subgroups (Kourovka 10.34, Monakhov)

1 fully proved1 ready to prove

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