Hall π-subgroups

Definitionπ-numbers (Kourovka 18.31)#✓fully proved

Let \(\pi\) be a set of natural numbers. A natural number \(n\) is a \(\pi\)-number when every prime divisor of \(n\) lies in \(\pi\).

def IsPiNumber (π : Set ℕ) (n : ℕ) : Prop :=
  ∀ p : ℕ, p.Prime → p ∣ n → p ∈ π
Used by 1
Used byπ-subgroups

Definitionπ-subgroups#✓fully proved

Let \(\pi\) be a set of natural numbers. A subgroup \(H\) of a group \(G\) is a \(\pi\)-subgroup when its order is a \(\pi\)-number, that is, every prime divisor of \(|H|\) lies in \(\pi\).

The order of an infinite subgroup is taken to be \(0\), which every prime divides.

def IsPiSubgroup {G : Type} [Group G] (π : Set ℕ) (H : Subgroup G) : Prop :=
  IsPiNumber π (Nat.card H)
Uses 1 · Used by 3
Statement usesπ-numbers (Kourovka 18.31)
Used byHall π-subgroups (Kourovka 18.31) · Maximal π-subgroups (Kourovka 18.31) · The π-core

DefinitionHall π-subgroups (Kourovka 18.31)#✓fully proved

Let \(\pi\) be a set of natural numbers. A subgroup \(H\) of a group \(G\) is a Hall \(\pi\)-subgroup when it is a \(\pi\)-subgroup and no prime divisor of its index \(|G : H|\) lies in \(\pi\).

An infinite index is taken to be \(0\), which every prime divides.

def IsHallPiSubgroup {G : Type} [Group G] (π : Set ℕ) (H : Subgroup G) : Prop :=
  IsPiSubgroup π H ∧ ∀ p : ℕ, p.Prime → p ∣ H.index → p ∉ π
Uses 1 · Used by 1
Statement usesπ-subgroups
Used byThree Hall π-subgroups meeting in the π-core (Kourovka 18.31)

DefinitionMaximal π-subgroups (Kourovka 18.31)#✓fully proved

Let \(\pi\) be a set of natural numbers. A subgroup \(H\) of a group \(G\) is a maximal \(\pi\)-subgroup when it is a \(\pi\)-subgroup and every \(\pi\)-subgroup \(K\) of \(G\) containing \(H\) equals \(H\).

def IsMaximalPiSubgroup {G : Type} [Group G] (π : Set ℕ) (H : Subgroup G) : Prop :=
  IsPiSubgroup π H ∧ ∀ K : Subgroup G, IsPiSubgroup π K → H ≤ K → K = H
Uses 1 · Used by 1
Statement usesπ-subgroups
Used byDπ-groups

DefinitionDπ-groups#✓fully proved

Let \(\pi\) be a set of natural numbers. A group \(G\) is a \(D_\pi\)-group when any two maximal \(\pi\)-subgroups \(H\) and \(K\) of \(G\) are conjugate, that is, \(K = gHg^{-1}\) for some \(g \in G\).

def IsDPiGroup (π : Set ℕ) (G : Type) [Group G] : Prop :=
  ∀ H K : Subgroup G, IsMaximalPiSubgroup π H → IsMaximalPiSubgroup π K →
    ∃ g : G, K = H.map (MulAut.conj g).toMonoidHom
Sources
  • Kourovka Archive, Problem 3.62.
Uses 1 · Used by 1
Statement usesMaximal π-subgroups (Kourovka 18.31)
Used byThree Hall π-subgroups meeting in the π-core (Kourovka 18.31)

DefinitionThe π-core#✓fully proved

Let \(\pi\) be a set of natural numbers. For a group \(G\), the \(\pi\)-core \(O_\pi(G)\) is the subgroup of \(G\) generated by all normal \(\pi\)-subgroups of \(G\).

When \(G\) is finite, \(O_\pi(G)\) is the largest normal \(\pi\)-subgroup of \(G\).

noncomputable def piCore (π : Set ℕ) (G : Type) [Group G] : Subgroup G :=
  sSup {N : Subgroup G | N.Normal ∧ IsPiSubgroup π N}
Uses 1 · Used by 1
Statement usesπ-subgroups
Used byThree Hall π-subgroups meeting in the π-core (Kourovka 18.31)

DefinitionThree Hall π-subgroups meeting in the π-core (Kourovka 18.31)#✓fully proved

For every set \(\pi\) of natural numbers and every finite \(D_\pi\)-group \(G\), there are Hall \(\pi\)-subgroups \(H_1\), \(H_2\), \(H_3\) of \(G\), not necessarily distinct, with \(H_1 \cap H_2 \cap H_3 = O_\pi(G)\), the \(\pi\)-core of \(G\).

This is the question of E. P. Vdovin and D. O. Revin; only the primes in \(\pi\) matter.

def threeHallSubgroupsIntersectInCore : Prop :=
  ∀ (π : Set ℕ) (G : Type) [Group G] [Finite G], IsDPiGroup π G →
    ∃ H₁ H₂ H₃ : Subgroup G, IsHallPiSubgroup π H₁ ∧ IsHallPiSubgroup π H₂ ∧
      IsHallPiSubgroup π H₃ ∧ H₁ ⊓ H₂ ⊓ H₃ = piCore π G
Uses 3 · Used by 1
Statement usesDπ-groups · Hall π-subgroups (Kourovka 18.31) · The π-core
Used byThree Hall π-subgroups need not meet in the π-core

TheoremThree Hall π-subgroups need not meet in the π-core#●ready to prove

It is not the case that for every set \(\pi\) of natural numbers every finite \(D_\pi\)-group \(G\) has Hall \(\pi\)-subgroups \(H_1\), \(H_2\), \(H_3\) with \(H_1 \cap H_2 \cap H_3 = O_\pi(G)\), the \(\pi\)-core of \(G\): the answer to Kourovka 18.31 is negative.

V. I. Zenkov (2022) gave a counterexample: \(G\) an extension of \(L_2(27)\) by a field automorphism of order \(3\), with a subgroup \(H\) that is an extension of a Borel subgroup of \(L_2(27)\) by a field automorphism of order \(3\).

public theorem_wanted threeHallSubgroupsIntersectInCore_refuted :
    ¬ threeHallSubgroupsIntersectInCore
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 18.31 (E. P. Vdovin, D. O. Revin); resolution: V. I. Zenkov, Siberian Math. J. 63 (2022), no. 4, 720-722, Kourovka Archive, Problem 18.31: No; G an extension of L_2(27) by a field automorphism of order 3, H an extension of a Borel subgroup of L_2(27) by a field automorphism of order 3, https://arxiv.org/abs/1401.0300v49.
Uses 1
Statement usesThree Hall π-subgroups meeting in the π-core (Kourovka 18.31)

Definition\(\pi\)-numbers#✓fully proved

Let \(\pi\) be a set of natural numbers. A natural number \(n\) is a \(\pi\)-number when every prime dividing \(n\) lies in \(\pi\).

def IsPiNumber (π : Set ℕ) (n : ℕ) : Prop :=
  ∀ p : ℕ, p.Prime → p ∣ n → p ∈ π
Used by 2
Used by\(D_\pi\)-groups · Hall \(\pi\)-subgroups

DefinitionHall \(\pi\)-subgroups#✓fully proved

Let \(\pi\) be a set of natural numbers. A subgroup \(H\) of a group \(G\) is a Hall \(\pi\)-subgroup when its order \(|H|\) is a \(\pi\)-number and no prime dividing its index \(|G : H|\) lies in \(\pi\). Here an infinite order or index is taken to be \(0\), which every prime divides.

def IsHallPiSubgroup (π : Set ℕ) {G : Type*} [Group G] (H : Subgroup G) : Prop :=
  IsPiNumber π (Nat.card H) ∧ ∀ p : ℕ, p.Prime → p ∣ H.index → p ∉ π
Uses 1 · Used by 1
Statement uses\(\pi\)-numbers
Used by\(D_\pi\)-groups

Definition\(D_\pi\)-groups#✓fully proved

Let \(\pi\) be a set of natural numbers (only the primes in \(\pi\) matter). A group \(G\) is a \(D_\pi\)-group (Kourovka Archive 3.62) when \(G\) has a Hall \(\pi\)-subgroup, any two Hall \(\pi\)-subgroups \(H\) and \(K\) of \(G\) are conjugate (\(K = gHg^{-1}\) for some \(g \in G\)), and every subgroup of \(G\) whose order is a \(\pi\)-number lies in some Hall \(\pi\)-subgroup. Here the order of an infinite subgroup is taken to be \(0\), which every prime divides.

def IsDPi (π : Set ℕ) (G : Type*) [Group G] : Prop :=
  (∃ H : Subgroup G, IsHallPiSubgroup π H) ∧
    (∀ H K : Subgroup G, IsHallPiSubgroup π H → IsHallPiSubgroup π K →
      ∃ g : G, K = H.map (MulAut.conj g).toMonoidHom) ∧
    ∀ P : Subgroup G, IsPiNumber π (Nat.card P) → ∃ H : Subgroup G, IsHallPiSubgroup π H ∧ P ≤ H
Uses 2 · Used by 1
Statement usesHall \(\pi\)-subgroups · \(\pi\)-numbers
Used byKourovka Problem 13.33

DefinitionKourovka Problem 13.33#✓fully proved

This is the proposition (F. Gross) that for every set \(\pi\) of natural numbers, every normal subgroup of a finite \(D_\pi\)-group is itself a \(D_\pi\)-group.

def normalSubgroupOfDPiIsDPi : Prop :=
  ∀ (π : Set ℕ) (G : Type) [Group G] [Finite G], IsDPi π G →
    ∀ N : Subgroup G, N.Normal → IsDPi π N
Uses 1 · Used by 1
Statement uses\(D_\pi\)-groups
Used byNormal subgroups of finite \(D_\pi\)-groups are \(D_\pi\)-groups

TheoremNormal subgroups of finite \(D_\pi\)-groups are \(D_\pi\)-groups#●ready to prove

Kourovka Problem 13.33 has a positive answer: for every set \(\pi\) of natural numbers, every normal subgroup of a finite \(D_\pi\)-group is itself a \(D_\pi\)-group.

The Kourovka Archive answers 13.33 "Yes, it is" modulo the classification of finite simple groups (E. P. Vdovin and D. O. Revin, 2006).

public theorem_wanted normalSubgroupOfDPiIsDPi_holds : normalSubgroupOfDPiIsDPi
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 13.33 (F. Gross; V. D. Mazurov); resolution: E. P. Vdovin and D. O. Revin, in: Ischia Group Theory 2004, Contemp. Math. 402, Amer. Math. Soc., 2006, 229-263, Kourovka Archive, Problem 13.33: Yes, it is mod CFSG.
Uses 1
Statement usesKourovka Problem 13.33

Definition\(\pi\)-number#✓fully proved

Let \(\pi\) be a set of natural numbers (only the primes in it matter). A natural number \(n\) is a \(\pi\)-number when every prime divisor of \(n\) lies in \(\pi\).

def IsPiNumber (π : Set ℕ) (n : ℕ) : Prop :=
  ∀ p : ℕ, p.Prime → p ∣ n → p ∈ π
Used by 2
Used byHall \(\pi\)-subgroup · Maximal \(\pi\)-subgroup

DefinitionHall \(\pi\)-subgroup#✓fully proved

Let \(\pi\) be a set of natural numbers and \(G\) a group. A subgroup \(H\) of \(G\) is a Hall \(\pi\)-subgroup when its order \(|H|\) is a \(\pi\)-number and no prime divisor of its index \(|G : H|\) lies in \(\pi\).

An infinite order or index counts as \(0\); the notion is meant for finite groups.

def IsHallPiSubgroup (π : Set ℕ) {G : Type*} [Group G] (H : Subgroup G) : Prop :=
  IsPiNumber π (Nat.card H) ∧ ∀ p : ℕ, p.Prime → p ∣ H.index → p ∉ π
Uses 1 · Used by 3
Statement uses\(\pi\)-number
Used byKourovka Problem 17.44(a): overgroups of Hall subgroups in \(C_\pi\)-groups · Kourovka Problem 17.44(b): overgroups of Hall subgroups in \(D_\pi\)-groups · \(C_\pi\)-group

DefinitionMaximal \(\pi\)-subgroup#✓fully proved

Let \(\pi\) be a set of natural numbers and \(G\) a group. A subgroup \(H\) of \(G\) is a maximal \(\pi\)-subgroup when its order \(|H|\) is a \(\pi\)-number and every subgroup \(L\) of \(G\) that contains \(H\) and whose order is a \(\pi\)-number equals \(H\).

An infinite order counts as \(0\); the notion is meant for finite groups.

def IsMaximalPiSubgroup (π : Set ℕ) {G : Type*} [Group G] (H : Subgroup G) : Prop :=
  IsPiNumber π (Nat.card H) ∧ ∀ L : Subgroup G, H ≤ L → IsPiNumber π (Nat.card L) → L = H
Uses 1 · Used by 1
Statement uses\(\pi\)-number
Used by\(D_\pi\)-group

Definition\(C_\pi\)-group#✓fully proved

Let \(\pi\) be a set of natural numbers. A group \(G\) is a \(C_\pi\)-group when it has a Hall \(\pi\)-subgroup and any two Hall \(\pi\)-subgroups \(H\) and \(K\) of \(G\) are conjugate, that is, \(K = gHg^{-1}\) for some \(g \in G\).

def IsCPiGroup (π : Set ℕ) (G : Type*) [Group G] : Prop :=
  (∃ H : Subgroup G, IsHallPiSubgroup π H) ∧ ∀ H K : Subgroup G, IsHallPiSubgroup π H →
    IsHallPiSubgroup π K → ∃ g : G, K = H.map (MulAut.conj g).toMonoidHom
Uses 1 · Used by 1
Statement usesHall \(\pi\)-subgroup
Used byKourovka Problem 17.44(a): overgroups of Hall subgroups in \(C_\pi\)-groups

Definition\(D_\pi\)-group#✓fully proved

Let \(\pi\) be a set of natural numbers. A group \(G\) is a \(D_\pi\)-group when any two maximal \(\pi\)-subgroups \(H\) and \(K\) of \(G\) are conjugate, that is, \(K = gHg^{-1}\) for some \(g \in G\).

def IsDPiGroup (π : Set ℕ) (G : Type*) [Group G] : Prop :=
  ∀ H K : Subgroup G, IsMaximalPiSubgroup π H → IsMaximalPiSubgroup π K →
    ∃ g : G, K = H.map (MulAut.conj g).toMonoidHom
Uses 1 · Used by 1
Statement usesMaximal \(\pi\)-subgroup
Used byKourovka Problem 17.44(b): overgroups of Hall subgroups in \(D_\pi\)-groups

DefinitionKourovka Problem 17.44(a): overgroups of Hall subgroups in \(C_\pi\)-groups#✓fully proved

The statement, posed by E. P. Vdovin and D. O. Revin, that for every set \(\pi\) of natural numbers and every finite \(C_\pi\)-group \(G\), every subgroup \(K\) of \(G\) that contains a Hall \(\pi\)-subgroup of \(G\) is itself a \(C_\pi\)-group.

def hallOvergroupCPi : Prop :=
  ∀ (π : Set ℕ) (G : Type) [Group G] [Finite G], IsCPiGroup π G →
    ∀ H K : Subgroup G, IsHallPiSubgroup π H → H ≤ K → IsCPiGroup π K
Uses 2 · Used by 1
Statement uses\(C_\pi\)-group · Hall \(\pi\)-subgroup
Used byOvergroups of Hall subgroups in \(C_\pi\)-groups (Vdovin–Revin)

DefinitionKourovka Problem 17.44(b): overgroups of Hall subgroups in \(D_\pi\)-groups#✓fully proved

The statement, posed by E. P. Vdovin and D. O. Revin, that for every set \(\pi\) of natural numbers and every finite \(D_\pi\)-group \(G\), every subgroup \(K\) of \(G\) that contains a Hall \(\pi\)-subgroup of \(G\) is itself a \(D_\pi\)-group.

def hallOvergroupDPi : Prop :=
  ∀ (π : Set ℕ) (G : Type) [Group G] [Finite G], IsDPiGroup π G →
    ∀ H K : Subgroup G, IsHallPiSubgroup π H → H ≤ K → IsDPiGroup π K
Uses 2 · Used by 1
Statement uses\(D_\pi\)-group · Hall \(\pi\)-subgroup
Used byOvergroups of Hall subgroups in \(D_\pi\)-groups (Manzaeva, Vdovin–Manzaeva–Revin)

TheoremOvergroups of Hall subgroups in \(C_\pi\)-groups (Vdovin–Revin)#●ready to prove

Let \(\pi\) be a set of natural numbers and let \(G\) be a finite \(C_\pi\)-group. Then every subgroup \(K\) of \(G\) that contains a Hall \(\pi\)-subgroup of \(G\) is itself a \(C_\pi\)-group. This answers Kourovka Problem 17.44(a) in the affirmative.

public theorem_wanted hallOvergroupCPi_holds : hallOvergroupCPi
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 17.44 (E. P. Vdovin, D. O. Revin); resolution: (a) E. P. Vdovin and D. O. Revin, Siberian Math. J. 54 (2013), 22-28; (b) N. Ch. Manzaeva, preprint, arXiv:1504.03137, and E. P. Vdovin, N. Ch. Manzaeva and D. O. Revin, Sb. Math. 211 (2020), 309-335, Kourovka Archive, Problem 17.44 (a) Yes, it is; b) Yes, it is), https://arxiv.org/abs/1504.03137.
Uses 1
Statement usesKourovka Problem 17.44(a): overgroups of Hall subgroups in \(C_\pi\)-groups

TheoremOvergroups of Hall subgroups in \(D_\pi\)-groups (Manzaeva, Vdovin–Manzaeva–Revin)#●ready to prove

Let \(\pi\) be a set of natural numbers and let \(G\) be a finite \(D_\pi\)-group. Then every subgroup \(K\) of \(G\) that contains a Hall \(\pi\)-subgroup of \(G\) is itself a \(D_\pi\)-group. This answers Kourovka Problem 17.44(b) in the affirmative.

public theorem_wanted hallOvergroupDPi_holds : hallOvergroupDPi
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 17.44 (E. P. Vdovin, D. O. Revin); resolution: (a) E. P. Vdovin and D. O. Revin, Siberian Math. J. 54 (2013), 22-28; (b) N. Ch. Manzaeva, preprint, arXiv:1504.03137, and E. P. Vdovin, N. Ch. Manzaeva and D. O. Revin, Sb. Math. 211 (2020), 309-335, Kourovka Archive, Problem 17.44 (a) Yes, it is; b) Yes, it is), https://arxiv.org/abs/1504.03137.
Uses 1
Statement usesKourovka Problem 17.44(b): overgroups of Hall subgroups in \(D_\pi\)-groups

Definitionπ-numbers#✓fully proved

Let \(\pi\) be a set of natural numbers. A natural number \(n\) is a \(\pi\)-number when every prime divisor of \(n\) lies in \(\pi\).

def IsPiNumber (π : Set ℕ) (n : ℕ) : Prop :=
  ∀ p : ℕ, p.Prime → p ∣ n → p ∈ π
Used by 1
Used byHall π-subgroups

DefinitionHall π-subgroups#✓fully proved

Let \(\pi\) be a set of natural numbers and let \(H\) be a subgroup of a group \(G\). Then \(H\) is a Hall \(\pi\)-subgroup when its order \(|H|\) is a \(\pi\)-number and no prime divisor of its index \(|G : H|\) lies in \(\pi\).

def IsHallPiSubgroup {G : Type} [Group G] (π : Set ℕ) (H : Subgroup G) : Prop :=
  IsPiNumber π (Nat.card H) ∧ ∀ p : ℕ, p.Prime → p ∣ H.index → p ∉ π
Uses 1 · Used by 1
Statement usesπ-numbers
Used byThe class \(E_\pi\) of groups with a Hall π-subgroup

DefinitionThe class \(E_\pi\) of groups with a Hall π-subgroup#✓fully proved

Let \(\pi\) be a set of natural numbers. A group \(G\) belongs to the class \(E_\pi\) when \(G\) has a Hall \(\pi\)-subgroup.

def HasHallPiSubgroup (π : Set ℕ) (G : Type) [Group G] : Prop :=
  ∃ H : Subgroup G, IsHallPiSubgroup π H
Uses 1 · Used by 1
Statement usesHall π-subgroups
Used byZavarnitsine's question on Hall subgroups for intersections of prime sets

DefinitionZavarnitsine's question on Hall subgroups for intersections of prime sets#✓fully proved

This is the statement that for all sets \(\pi_1\) and \(\pi_2\) of natural numbers, every finite group that has a Hall \(\pi_1\)-subgroup and a Hall \(\pi_2\)-subgroup also has a Hall \((\pi_1 \cap \pi_2)\)-subgroup; that is, writing \(E_\pi\) for the class of finite groups with a Hall \(\pi\)-subgroup, \(E_{\pi_1} \cap E_{\pi_2}\) is contained in \(E_{\pi_1 \cap \pi_2}\).

This is Kourovka Notebook Problem 18.38 (A. V. Zavarnitsine).

def hallSubgroupForIntersectionOfPrimeSets : Prop :=
  ∀ (π₁ π₂ : Set ℕ) (G : Type) [Group G] [Finite G],
    HasHallPiSubgroup π₁ G → HasHallPiSubgroup π₂ G → HasHallPiSubgroup (π₁ ∩ π₂) G
Uses 1 · Used by 1
Statement usesThe class \(E_\pi\) of groups with a Hall π-subgroup
Used byHall subgroups exist for intersections of prime sets

TheoremHall subgroups exist for intersections of prime sets#●ready to prove

For all sets \(\pi_1\) and \(\pi_2\) of natural numbers, every finite group that has a Hall \(\pi_1\)-subgroup and a Hall \(\pi_2\)-subgroup also has a Hall \((\pi_1 \cap \pi_2)\)-subgroup; that is, writing \(E_\pi\) for the class of finite groups with a Hall \(\pi\)-subgroup, \(E_{\pi_1} \cap E_{\pi_2}\) is contained in \(E_{\pi_1 \cap \pi_2}\).

This answers Kourovka Notebook Problem 18.38 (A. V. Zavarnitsine) in the affirmative; the 21st issue records the answer "Yes, it does" (N. Yang and A. A. Buturlakin, 2025).

public theorem_wanted hallSubgroupForIntersectionOfPrimeSets_holds :
    hallSubgroupForIntersectionOfPrimeSets
Sources
  • E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.38 (A. V. Zavarnitsine); resolution: N. Yang and A. A. Buturlakin, A generalization of the Arad-Ward theorem on Hall subgroups, preprint, arXiv:2501.05865 (2025), abstract: E_{π₁} ∩ E_{π₂} is contained in E_{π₁ ∩ π₂}; the Kourovka Notebook (21st issue) marks 18.38 solved, https://arxiv.org/abs/2501.05865.
Uses 1
Statement usesZavarnitsine's question on Hall subgroups for intersections of prime sets
22 fully proved5 ready to prove

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