Conjugacy of maximal subgroups

DefinitionSchmid's question on isomorphic simple maximal subgroups (Kourovka 18.115)#✓fully proved

The statement that in every finite simple group \(G\), any two isomorphic maximal subgroups \(X\) and \(Y\) that are themselves simple groups are conjugate in \(\operatorname{Aut} G\), that is, \(\varphi(X) = Y\) for some automorphism \(\varphi\) of \(G\).

def isomorphicSimpleMaximalSubgroupsAutConjugate : Prop :=
  ∀ (G : Type) [Group G] [Finite G], IsSimpleGroup G →
    ∀ X Y : Subgroup G, IsCoatom X → IsCoatom Y → IsSimpleGroup X → IsSimpleGroup Y →
      Nonempty (X ≃* Y) → ∃ φ : MulAut G, X.map φ.toMonoidHom = Y

Used by 1

Used byIsomorphic simple maximal subgroups need not be conjugate under automorphisms

TheoremIsomorphic simple maximal subgroups need not be conjugate under automorphisms#●ready to prove

There are a finite simple group \(G\) and isomorphic maximal subgroups \(X\) and \(Y\) of \(G\), both simple groups, such that no automorphism \(\varphi\) of \(G\) satisfies \(\varphi(X) = Y\). That is, the answer to Kourovka Problem 18.115 is negative.

The Kourovka Archive answers 18.115 "No, not always": there are two \(\operatorname{Aut} G\)-classes of \(M_{12}\) in \(E_6(5)\) (Kleidman–Wilson, 1990), and Craven (2023) found two classes of \(\mathrm{PSL}_2(11)\) and two of \(\mathrm{PSL}_2(19)\) in \(E_6(q)\) for certain \(q\).

public theorem_wanted isomorphicSimpleMaximalSubgroupsAutConjugate_refuted :
    ¬ isomorphicSimpleMaximalSubgroupsAutConjugate

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 18.115 (P. Schmid); resolution: P. B. Kleidman and R. A. Wilson, J. London Math. Soc. (2) 42 (1990), 555-561; D. A. Craven, Invent. Math. 234 (2023), no. 2, 637-719, Kourovka Archive, Problem 18.115: No, not always; two Aut G-classes of M_12 in E_6(5) (Kleidman-Wilson), and two classes of PSL_2(11) and of PSL_2(19) in E_6(q) for certain q (Craven), https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesSchmid's question on isomorphic simple maximal subgroups (Kourovka 18.115)

1 fully proved1 ready to prove

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