Residually finite p-groups and residual nilpotence

DefinitionKuz'min's question on residually finite \(p\)-groups (Kourovka 14.52)#✓fully proved

The statement that every finitely generated group \(G\) that is residually a finite \(p\)-group for every prime \(p\) is residually torsion-free nilpotent. Here \(G\) is residually a finite \(p\)-group when every element \(g \ne 1\) lies outside some normal subgroup \(N\) such that \(G/N\) is a finite \(p\)-group, and residually torsion-free nilpotent when every element \(g \ne 1\) lies outside some normal subgroup \(N\) such that \(G/N\) is nilpotent and has no nontrivial elements of finite order.

def residuallyFinitePImpliesResiduallyTorsionFreeNilpotent : Prop :=
  ∀ (G : Type) [Group G], Group.FG G →
    (∀ p : ℕ, p.Prime → ∀ g : G, g ≠ 1 → ∃ (N : Subgroup G) (_ : N.Normal), g ∉ N ∧
      Finite (G ⧸ N) ∧ IsPGroup p (G ⧸ N)) →
    ∀ g : G, g ≠ 1 → ∃ (N : Subgroup G) (_ : N.Normal), g ∉ N ∧ Group.IsNilpotent (G ⧸ N) ∧
      ∀ y : G ⧸ N, y ≠ 1 → ¬ IsOfFinOrder y

Used by 1

Used byResidual \(p\)-finiteness for all \(p\) does not imply residual torsion-free nilpotence

TheoremResidual \(p\)-finiteness for all \(p\) does not imply residual torsion-free nilpotence#●ready to prove

It is not the case that every finitely generated group that is residually a finite \(p\)-group for every prime \(p\) is residually torsion-free nilpotent. Here a group \(G\) is residually a finite \(p\)-group when every element \(g \ne 1\) lies outside some normal subgroup \(N\) such that \(G/N\) is a finite \(p\)-group, and residually torsion-free nilpotent when every element \(g \ne 1\) lies outside some normal subgroup \(N\) such that \(G/N\) is nilpotent and has no nontrivial elements of finite order.

public theorem_wanted residuallyFinitePImpliesResiduallyTorsionFreeNilpotent_refuted :
    ¬ residuallyFinitePImpliesResiduallyTorsionFreeNilpotent

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 14.52 (Yu. V. Kuz'min); resolution: B. Hartley, in: Symposia Mathematica XVII (Convegno sui Gruppi Infiniti, INDAM, Roma, 1973), Academic Press, London, 1976, 225-234, Kourovka Archive, Problem 14.52 (answer: Not always).

Uses 1

Statement usesKuz'min's question on residually finite \(p\)-groups (Kourovka 14.52)

1 fully proved1 ready to prove

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