2-groups in which normal closures determine conjugacy

DefinitionGolikova–Starostin question on finite \(2\)-groups in which normal closures determine conjugacy (Kourovka 12.15)#✓fully proved

Golikova–Starostin question on finite \(2\)-groups in which normal closures determine conjugacy (Kourovka 12.15). The statement that for every finite \(2\)-group \(G\) in which any two elements with the same normal closure are conjugate, the derived subgroup \(G'\) of \(G\) is abelian.

def normalClosureConjDerivedAbelian : Prop :=
  ∀ (G : Type) [Group G] [Finite G], IsPGroup 2 G →
    (∀ x y : G, Subgroup.normalClosure {x} = Subgroup.normalClosure {y} → IsConj x y) →
    ∀ a ∈ commutator G, ∀ b ∈ commutator G, a * b = b * a

Used by 1

Used byEqual normal closures forcing conjugacy does not make the derived subgroup of a finite \(2\)-group abelian (Kourovka 12.15)

TheoremEqual normal closures forcing conjugacy does not make the derived subgroup of a finite \(2\)-group abelian (Kourovka 12.15)#●ready to prove

Equal normal closures forcing conjugacy does not make the derived subgroup of a finite \(2\)-group abelian (Kourovka 12.15). There exists a finite \(2\)-group \(G\) in which any two elements with the same normal closure are conjugate, but whose derived subgroup \(G'\) is not abelian.

Muliarchyk gives an explicit such group of order \(2^{11} = 2048\), given by a power-commutator presentation, whose second derived subgroup \(G'' = (G')'\) is nontrivial (a ChatGPT-assisted construction with a symbolic proof); the Kourovka Notebook (21st issue) marks 12.15 solved, "No, not always".

public theorem_wanted normalClosureConjDerivedAbelian_refuted : ¬ normalClosureConjDerivedAbelian

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 12.15 (E. A. Golikova, A. I. Starostin); resolution: K. Muliarchyk, An explicit counterexample to Kourovka Notebook Problem 12.15, preprint (13 August 2026), main result: an explicit group of order 2¹¹ = 2048, given by a power-commutator presentation, satisfies the hypothesis and has G'' = ⟨z⟩ ≠ 1; the Kourovka Notebook (21st issue) marks 12.15 solved, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/08/kourovka_12_15_counterexample.pdf.

Uses 1

Statement usesGolikova–Starostin question on finite \(2\)-groups in which normal closures determine conjugacy (Kourovka 12.15)

1 fully proved1 ready to prove

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