Pronormality of Hall subgroups

DefinitionHall subgroups (order coprime to index)#✓fully proved

A subgroup \(H\) of a group \(G\) is a Hall subgroup when its order \(|H|\) is coprime to its index \(|G : H|\).

def IsHallSubgroup {G : Type} [Group G] (H : Subgroup G) : Prop :=
  (Nat.card H).Coprime H.index

Used by 1

Used byVdovin–Revin question on Hall subgroups and their normal closures

DefinitionPronormality in a subgroup#✓fully proved

Let \(H\) and \(K\) be subgroups of a group \(G\). Then \(H\) is pronormal in \(K\) when for every \(g \in K\) there is an element \(x\) of the subgroup \(\langle H, gHg^{-1} \rangle\) generated by \(H\) and \(gHg^{-1}\) such that \(xHx^{-1} = gHg^{-1}\), that is, \(H\) and its conjugate \(gHg^{-1}\) are conjugate by an element of \(\langle H, gHg^{-1} \rangle\).

def IsPronormalIn {G : Type} [Group G] (H K : Subgroup G) : Prop :=
  ∀ g ∈ K, ∃ x ∈ H ⊔ H.map (MulAut.conj g).toMonoidHom,
    H.map (MulAut.conj x).toMonoidHom = H.map (MulAut.conj g).toMonoidHom

Used by 1

Used byVdovin–Revin question on Hall subgroups and their normal closures

DefinitionVdovin–Revin question on Hall subgroups and their normal closures#✓fully proved

This is the statement that for every finite group \(G\) and every Hall subgroup \(H\) of \(G\) (a subgroup whose order \(|H|\) is coprime to its index \(|G : H|\)), \(H\) is pronormal in its normal closure \(\langle H^G \rangle\), the smallest normal subgroup of \(G\) containing \(H\): for every \(g \in \langle H^G \rangle\) there is an element \(x\) of the subgroup \(\langle H, gHg^{-1} \rangle\) generated by \(H\) and \(gHg^{-1}\) with \(xHx^{-1} = gHg^{-1}\).

This is Kourovka Notebook Problem 18.32 (E. P. Vdovin, D. O. Revin).

def hallSubgroupPronormalInNormalClosure : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (H : Subgroup G), IsHallSubgroup H →
    IsPronormalIn H (Subgroup.normalClosure (H : Set G))

Uses 2 · Used by 1

Statement usesHall subgroups (order coprime to index) · Pronormality in a subgroup

Used byHall subgroups need not be pronormal in their normal closure

TheoremHall subgroups need not be pronormal in their normal closure#●ready to prove

There exist a finite group \(G\) and a Hall subgroup \(H\) of \(G\) (that is, the order \(|H|\) is coprime to the index \(|G : H|\)) such that \(H\) is not pronormal in its normal closure \(\langle H^G \rangle\), the smallest normal subgroup of \(G\) containing \(H\): for some \(g \in \langle H^G \rangle\), no element \(x\) of the subgroup \(\langle H, gHg^{-1} \rangle\) generated by \(H\) and \(gHg^{-1}\) satisfies \(xHx^{-1} = gHg^{-1}\).

This answers Kourovka Notebook Problem 18.32 (E. P. Vdovin, D. O. Revin) in the negative; the Kourovka Archive records the answer "No, not always" (M. N. Nesterov, 2017).

public theorem_wanted hallSubgroupPronormalInNormalClosure_refuted :
    ¬ hallSubgroupPronormalInNormalClosure

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 18.32 (E. P. Vdovin, D. O. Revin); resolution: M. N. Nesterov, Siberian Math. J. 58 (2017), no. 1, 128-133, Kourovka Archive, Problem 18.32: No, not always, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesVdovin–Revin question on Hall subgroups and their normal closures

3 fully proved1 ready to prove

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