Propermutable subgroups

DefinitionPermuting subgroups#✓fully proved

Let \(H\) and \(X\) be subgroups of a group \(G\). Then \(H\) permutes with \(X\) when \(HX = XH\) as subsets of \(G\), where \(HX = \{hx : h \in H, x \in X\}\) and \(XH = \{xh : x \in X, h \in H\}\).

def Permutes {G : Type} [Group G] (H X : Subgroup G) : Prop :=
  (H : Set G) * (X : Set G) = (X : Set G) * (H : Set G)

Used by 1

Used byPropermutable subgroups

DefinitionPropermutable subgroups#✓fully proved

A subgroup \(H\) of a group \(G\) is propermutable in \(G\) when there is a subgroup \(B\) of \(G\) with \(G = N_G(H)B\) (every element of \(G\) is a product \(nb\) with \(n\) in the normalizer \(N_G(H)\) of \(H\) and \(b \in B\)) such that \(H\) permutes with every subgroup \(X\) of \(B\), that is, \(HX = XH\).

def IsPropermutable {G : Type} [Group G] (H : Subgroup G) : Prop :=
  ∃ B : Subgroup G, (Subgroup.normalizer (H : Set G) : Set G) * (B : Set G) = Set.univ ∧
    ∀ X : Subgroup G, X ≤ B → Permutes H X

Uses 1 · Used by 2

Statement usesPermuting subgroups

Used byKourovka Problem 18.91(a) (statement) · Kourovka Problem 18.91(b) (statement)

Definition2-nilpotent groups#✓fully proved

A group \(G\) is \(2\)-nilpotent when it has a normal subgroup \(N\) of finite odd order whose index in \(G\) is \(2^k\) for some natural number \(k\).

def IsTwoNilpotent (G : Type) [Group G] : Prop :=
  ∃ N : Subgroup G, N.Normal ∧ Odd (Nat.card N) ∧ ∃ k : ℕ, N.index = 2 ^ k

Used by 1

Used byKourovka Problem 18.91(b) (statement)

DefinitionKourovka Problem 18.91(a) (statement)#✓fully proved

The statement of Kourovka Problem 18.91(a) (A. N. Skiba): there exist a finite group \(G\) and subgroups \(A \le B \le G\) such that \(A\) is propermutable in \(G\) but \(A\), regarded as a subgroup of \(B\), is not propermutable in \(B\).

def propermutableNotInIntermediate : Prop :=
  ∃ (G : Type) (_ : Group G) (_ : Finite G) (A B : Subgroup G),
    A ≤ B ∧ IsPropermutable A ∧ ¬ IsPropermutable (A.subgroupOf B)

Uses 1 · Used by 1

Statement usesPropermutable subgroups

Used byPropermutability is not inherited by intermediate subgroups

DefinitionKourovka Problem 18.91(b) (statement)#✓fully proved

The statement of Kourovka Problem 18.91(b) (A. N. Skiba), with the range \(1 \le k < n\): let \(G\) be a finite group and \(P\) a non-abelian Sylow \(2\)-subgroup of \(G\), with \(|P| = 2^n\), and let \(k\) be a natural number with \(1 \le k < n\). If every subgroup of \(P\) of order \(2^k\) is propermutable in \(G\), and, when \(k = 1\), every cyclic subgroup of \(P\) of order \(4\) is propermutable in \(G\), then \(G\) is \(2\)-nilpotent.

The printed problem has \(1 < k < n\), which contradicts its clause for \(k = 1\); the range \(1 \le k < n\) is the one of Al-Sharo's Corollary 1.3.

def propermutableSylowSubgroupsTwoNilpotent : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (P : Sylow 2 G),
    (∃ x ∈ (P : Subgroup G), ∃ y ∈ (P : Subgroup G), x * y ≠ y * x) →
    ∀ k : ℕ, 1 ≤ k → 2 ^ k < Nat.card (P : Subgroup G) →
      (∀ H : Subgroup G, H ≤ P → Nat.card H = 2 ^ k → IsPropermutable H) →
      (k = 1 → ∀ H : Subgroup G, H ≤ P → IsCyclic H → Nat.card H = 4 → IsPropermutable H) →
      IsTwoNilpotent G

Uses 2 · Used by 1

Statement usesPropermutable subgroups · 2-nilpotent groups

Used byPropermutable subgroups of a non-abelian Sylow 2-subgroup force 2-nilpotency

TheoremPropermutability is not inherited by intermediate subgroups#●ready to prove

There exist a finite group \(G\) and subgroups \(A \le B \le G\) such that \(A\) is propermutable in \(G\) but \(A\), regarded as a subgroup of \(B\), is not propermutable in \(B\). This is the positive answer to Kourovka Problem 18.91(a).

The Kourovka Archive answers 18.91(a) "Yes, there is" (A. A. Pypka and D. Yu. Storozhenko, 2017).

public theorem_wanted propermutableNotInIntermediate_holds : propermutableNotInIntermediate

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 18.91 (A. N. Skiba); resolution: (a) A. A. Pypka and D. Yu. Storozhenko, Dopov. Nac. Akad. Nauk Ukrain. 2017, no. 7, 18-20 (Ukrainian); (b) Kh. A. Al-Sharo, Finite groups with given systems of weakly S-propermutable subgroups, J. Group Theory 19 (2016), no. 5, 871-887, Kourovka Archive, Problem 18.91: a) Yes, there is; b) Yes, it is. Al-Sharo, Corollary 1.3: if X ⊴ G, P ∈ Syl_p(X), (p - 1, |X|) = 1, and for some 1 < D < P every subgroup of P of order |D| (and, when P is non-abelian and |D| = 2, every cyclic subgroup of order 4) is S-propermutable in G, then X is p-nilpotent, https://doi.org/10.1515/jgth-2016-0017.

Uses 1

Statement usesKourovka Problem 18.91(a) (statement)

TheoremPropermutable subgroups of a non-abelian Sylow 2-subgroup force 2-nilpotency#●ready to prove

Let \(G\) be a finite group and \(P\) a non-abelian Sylow \(2\)-subgroup of \(G\), with \(|P| = 2^n\), and let \(k\) be a natural number with \(1 \le k < n\). If every subgroup of \(P\) of order \(2^k\) is propermutable in \(G\), and, when \(k = 1\), every cyclic subgroup of \(P\) of order \(4\) is propermutable in \(G\), then \(G\) is \(2\)-nilpotent. This is the positive answer to Kourovka Problem 18.91(b).

The Kourovka Archive answers 18.91(b) "Yes, it is" (Kh. A. Al-Sharo, 2016). Al-Sharo's Corollary 1.3 (quoted below), applied to the normal subgroup \(X = G\) with \(p = 2\) (where \((p - 1, |G|) = 1\) holds), gives 2-nilpotency already when the subgroups in question are only \(S\)-propermutable, a weaker condition than propermutability: a subgroup \(H\) of \(G\) is \(S\)-propermutable in \(G\) when there is a subgroup \(B\) of \(G\) with \(G = N_G(H)B\) such that \(H\) permutes with every Sylow subgroup of \(B\).

public theorem_wanted propermutableSylowSubgroupsTwoNilpotent_holds :
    propermutableSylowSubgroupsTwoNilpotent

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 18.91 (A. N. Skiba); resolution: (a) A. A. Pypka and D. Yu. Storozhenko, Dopov. Nac. Akad. Nauk Ukrain. 2017, no. 7, 18-20 (Ukrainian); (b) Kh. A. Al-Sharo, Finite groups with given systems of weakly S-propermutable subgroups, J. Group Theory 19 (2016), no. 5, 871-887, Kourovka Archive, Problem 18.91: a) Yes, there is; b) Yes, it is. Al-Sharo, Corollary 1.3: if X ⊴ G, P ∈ Syl_p(X), (p - 1, |X|) = 1, and for some 1 < D < P every subgroup of P of order |D| (and, when P is non-abelian and |D| = 2, every cyclic subgroup of order 4) is S-propermutable in G, then X is p-nilpotent, https://doi.org/10.1515/jgth-2016-0017.

Uses 1

Statement usesKourovka Problem 18.91(b) (statement)

5 fully proved2 ready to prove

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