Polynormal and paranormal subgroups

DefinitionConjugate subgroup#✓fully proved

For a subgroup \(A\) of a group \(G\) and an element \(x \in G\), the conjugate subgroup is \(A^x = x^{-1} A x = \{x^{-1} a x : a \in A\}\).

def conjSubgroup (A : Subgroup G) (x : G) : Subgroup G :=
  A.map (MulAut.conj x⁻¹).toMonoidHom

Used by 2

Used byParanormal subgroup · Polynormal subgroup

DefinitionParanormal subgroup#✓fully proved

A subgroup \(A\) of a group \(G\) is paranormal (Z. I. Borevich) when, for every \(x \in G\), the conjugate \(A^x = x^{-1} A x\) is contained in the subgroup generated by the conjugates \(A^u\) with \(u\) in the subgroup \(\langle A, A^x \rangle\) generated by \(A\) and \(A^x\).

def IsParanormal (A : Subgroup G) : Prop :=
  ∀ x : G, conjSubgroup A x ≤ ⨆ u ∈ A ⊔ conjSubgroup A x, conjSubgroup A u

Uses 1 · Used by 1

Statement usesConjugate subgroup

Used byPolynormal subgroups of finite groups are paranormal

DefinitionPolynormal subgroup#✓fully proved

A subgroup \(A\) of a group \(G\) is polynormal (Z. I. Borevich) when, for every \(x \in G\), the conjugate \(A^x = x^{-1} A x\) is contained in the subgroup generated by the conjugates \(A^u\) with \(u \in A^{\langle x \rangle}\), where \(A^{\langle x \rangle}\) is the subgroup generated by the conjugates \(A^{x^n}\) with \(n \in \mathbb{Z}\).

def IsPolynormal (A : Subgroup G) : Prop :=
  ∀ x : G, conjSubgroup A x ≤ ⨆ u ∈ (⨆ n : ℤ, conjSubgroup A (x ^ n)), conjSubgroup A u

Uses 1 · Used by 1

Statement usesConjugate subgroup

Used byPolynormal subgroups of finite groups are paranormal

DefinitionPolynormal subgroups of finite groups are paranormal#✓fully proved

Kourovka 14.50 (Z. I. Borevich; A. S. Kondratiev): every polynormal subgroup of a finite group is paranormal.

def polynormalImpliesParanormal : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (A : Subgroup G), IsPolynormal A → IsParanormal A

Uses 2 · Used by 1

Statement usesParanormal subgroup · Polynormal subgroup

Used byA polynormal subgroup that is not paranormal (Kourovka 14.50 refuted)

TheoremA polynormal subgroup that is not paranormal (Kourovka 14.50 refuted)#●ready to prove

There is a finite group \(G\) with a subgroup \(A\) that is polynormal but not paranormal. So the conjecture of Kourovka Problem 14.50 is false.

public theorem_wanted polynormalImpliesParanormal_refuted : ¬ polynormalImpliesParanormal

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 14.50 (Z. I. Borevich; A. S. Kondratiev); resolution: V. I. Mysovskikh, Dokl. Math. 60 (1999), 71-72, Kourovka Archive, Problem 14.50 (answer: Not always).

Uses 1

Statement usesPolynormal subgroups of finite groups are paranormal

4 fully proved1 ready to prove

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