Derived subgroups of subgroups of free groups

DefinitionInclusion of derived subgroups in free groups (Kourovka 13.62)#✓fully proved

The conjecture of V. P. Shaptala that for every free group \(F\) and all non-cyclic subgroups \(U\) and \(V\) of \(F\), if the commutator subgroup \([U, U]\) is contained in the commutator subgroup \([V, V]\), then \(U\) is contained in \(V\).

def derivedInclusionImpliesInclusion : Prop :=
  ∀ (α : Type) (U V : Subgroup (FreeGroup α)), ¬ IsCyclic U → ¬ IsCyclic V →
    ⁅U, U⁆ ≤ ⁅V, V⁆ → U ≤ V

Used by 1

Used byInclusion of derived subgroups does not imply inclusion in free groups

TheoremInclusion of derived subgroups does not imply inclusion in free groups#●ready to prove

There exist a free group \(F\) and non-cyclic subgroups \(U\) and \(V\) of \(F\) such that the commutator subgroup \([U, U]\) is contained in the commutator subgroup \([V, V]\) but \(U\) is not contained in \(V\). That is, the conjecture of Kourovka Problem 13.62 is false; the Kourovka Archive credits this answer to M. J. Dunwoody.

public theorem_wanted derivedInclusionImpliesInclusion_refuted : ¬ derivedInclusionImpliesInclusion

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 13.62 (V. P. Shaptala); resolution: M. J. Dunwoody, Arch. Math. 16 (1965), 153-157, Kourovka Archive, Problem 13.62 (answer: No, it does not).

Uses 1

Statement usesInclusion of derived subgroups in free groups (Kourovka 13.62)

1 fully proved1 ready to prove

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