Permutable subgroups of alternating groups

Definition\(G\)-permutable subgroups#✓fully proved

A subgroup \(A\) of a group \(G\) is \(G\)-permutable when for every subgroup \(B\) of \(G\) there is \(x \in G\) such that \(A B^x = B^x A\), where \(B^x = x B x^{-1}\) is the conjugate of \(B\) by \(x\) and the products are products of subsets of \(G\).

def IsGPermutable {G : Type*} [Group G] (A : Subgroup G) : Prop :=
  ∀ B : Subgroup G, ∃ x : G,
    (A : Set G) * (B.map (MulAut.conj x).toMonoidHom : Set G) =
      (B.map (MulAut.conj x).toMonoidHom : Set G) * (A : Set G)

Used by 1

Used byKourovka Problem 17.37 on permutable subgroups of alternating groups

DefinitionKourovka Problem 17.37 on permutable subgroups of alternating groups#✓fully proved

This proposition asserts that there is a natural number \(n\) such that for every \(m > n\), every \(A_m\)-permutable subgroup of the alternating group \(A_m\) is either the trivial subgroup or \(A_m\) itself. Here a subgroup \(A\) of a group \(G\) is \(G\)-permutable when for every subgroup \(B\) of \(G\) there is \(x \in G\) with \(A B^x = B^x A\), where \(B^x = x B x^{-1}\) is the conjugate of \(B\) by \(x\) and products of subgroups are products of subsets of \(G\).

The trivial subgroup and \(A_m\) itself are always \(A_m\)-permutable.

def alternatingGroupNoProperPermutableSubgroups : Prop :=
  ∃ n : ℕ, ∀ m : ℕ, n < m → ∀ A : Subgroup (alternatingGroup (Fin m)),
    IsGPermutable A → A = ⊥ ∨ A = ⊤

Uses 1 · Used by 1

Statement uses\(G\)-permutable subgroups

Used byPermutable subgroups of large alternating groups are trivial or the whole group

TheoremPermutable subgroups of large alternating groups are trivial or the whole group#●ready to prove

There is a natural number \(n\) such that for every \(m > n\), every \(A_m\)-permutable subgroup of the alternating group \(A_m\) is either the trivial subgroup or \(A_m\) itself. Here a subgroup \(A\) of a group \(G\) is \(G\)-permutable when for every subgroup \(B\) of \(G\) there is \(x \in G\) with \(A B^x = B^x A\), where \(B^x = x B x^{-1}\).

public theorem_wanted alternatingGroupNoProperPermutableSubgroups_holds :
    alternatingGroupNoProperPermutableSubgroups

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 17.37 (A. F. Vasil'ev, A. N. Skiba); resolution: N. Yang and A. Galt, Bull. Malays. Math. Sci. Soc. (2) 46 (2023), no. 5, Paper no. 177, as stated in S. Zhang, Ordinary G-permutable subgroups of finite simple groups, preprint, arXiv:2610.12329 (2026), Zhang, Introduction (Yang and Galt proved that there is N ≥ 7 such that no A_n with n ≥ N has a proper nontrivial A_n-permutable subgroup, with A_n-permutable as in Problem 17.112); the Kourovka Notebook (21st issue) marks 17.37 solved (Yes, there is), https://arxiv.org/abs/2610.12329.

Uses 1

Statement usesKourovka Problem 17.37 on permutable subgroups of alternating groups

2 fully proved1 ready to prove

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