Nilpotent pairs in finite subsets of groups

DefinitionKourovka Problem 17.3 (Abdollahi's question on nilpotent pairs)#✓fully proved

The proposition that for every group \(G\) in which every \(4\)-element subset contains two distinct elements generating a nilpotent subgroup, every subgroup of \(G\) generated by two elements is nilpotent.

def fourSubsetNilpotentPairTwoGeneratedNilpotent : Prop :=
  ∀ (G : Type) [Group G],
    (∀ s : Finset G, s.card = 4 →
      ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ Group.IsNilpotent (Subgroup.closure {x, y})) →
    ∀ x y : G, Group.IsNilpotent (Subgroup.closure {x, y})

Used by 1

Used byKourovka Problem 17.3 has a negative answer

TheoremKourovka Problem 17.3 has a negative answer#●ready to prove

There is a group \(G\) in which every \(4\)-element subset contains two distinct elements generating a nilpotent subgroup, but some subgroup of \(G\) generated by two elements is not nilpotent.

The Kourovka Archive answers 17.3 "No, not always", citing Sozutov (2019).

public theorem_wanted fourSubsetNilpotentPairTwoGeneratedNilpotent_refuted :
    ¬ fourSubsetNilpotentPairTwoGeneratedNilpotent

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 17.3 (A. Abdollahi); resolution: A. I. Sozutov, Siberian Math. J. 60 (2019), 1099-1100, Kourovka Archive, Problem 17.3 (No, not always).

Uses 1

Statement usesKourovka Problem 17.3 (Abdollahi's question on nilpotent pairs)

1 fully proved1 ready to prove

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