Intersection of the derived series

DefinitionPerfectness of the intersection of the derived series (Kourovka 17.82)#✓fully proved

The conjecture of R. Mikhailov that for every finitely presented group \(G\), the intersection \(D = \bigcap_{n} G^{(n)}\) of the terms \(G^{(n)}\) of the derived series of \(G\) satisfies \([D, D] = D\), that is, \(D\) has trivial abelianization.

def derivedSeriesIntersectionPerfect : Prop :=
  ∀ (G : Type) [Group G], Group.IsFinitelyPresented G →
    ⁅⨅ n, derivedSeries G n, ⨅ n, derivedSeries G n⁆ = ⨅ n, derivedSeries G n

Used by 1

Used byThe intersection of the derived series need not be perfect

TheoremThe intersection of the derived series need not be perfect#●ready to prove

There exists a finitely presented group \(G\) such that the intersection \(D = \bigcap_{n} G^{(n)}\) of the terms \(G^{(n)}\) of the derived series of \(G\) satisfies \([D, D] \ne D\). That is, the conjecture of Kourovka Problem 17.82 is false; the Kourovka Archive credits this answer to V. A. Roman'kov (2011).

public theorem_wanted derivedSeriesIntersectionPerfect_refuted : ¬ derivedSeriesIntersectionPerfect

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 17.82 (R. Mikhailov); resolution: V. A. Roman'kov, Siberian Math. J. 52 (2011), 348-351, Kourovka Archive, Problem 17.82 (No, not always).

Uses 1

Statement usesPerfectness of the intersection of the derived series (Kourovka 17.82)

1 fully proved1 ready to prove

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