Commutator power identities

DefinitionDerived subgroups under a commutator power identity (Kourovka 13.34)#✓fully proved

This is the proposition that for every integer \(n \ge 1\) and every group \(G\) with \([x, y]^n = 1\) for all \(x, y \in G\), every element of the derived subgroup \([G, G]\) has finite order, that is, the derived subgroup of \(G\) is periodic.

This is Kourovka Notebook Problem 13.34 (I. D. Macdonald).

def commutatorIdentityPeriodicDerived : Prop :=
  ∀ n : ℕ, 1 ≤ n → ∀ (G : Type) [Group G], (∀ x y : G, ⁅x, y⁆ ^ n = 1) →
    ∀ g ∈ commutator G, IsOfFinOrder g

Used by 1

Used byA commutator power identity need not make the derived subgroup periodic

TheoremA commutator power identity need not make the derived subgroup periodic#●ready to prove

There exist an integer \(n \ge 1\) and a group \(G\) such that \([x, y]^n = 1\) for all \(x, y \in G\) but some element of the derived subgroup \([G, G]\) has infinite order. This answers Kourovka Notebook Problem 13.34 (I. D. Macdonald; V. D. Mazurov) in the negative, by results of G. S. Deryabina and P. A. Kozhevnikov from 1999 and of S. I. Adyan from 2000.

public theorem_wanted commutatorIdentityPeriodicDerived_refuted :
    ¬ commutatorIdentityPeriodicDerived

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 13.34 (I. D. Macdonald; V. D. Mazurov); resolution: G. S. Deryabina and P. A. Kozhevnikov, Commun. Algebra 27 (1999), no. 9, 4525-4530; S. I. Adyan, Dokl. Math. 62 (2000), no. 2, 174-176, Kourovka Archive, Problem 13.34: No, not always.

Uses 1

Statement usesDerived subgroups under a commutator power identity (Kourovka 13.34)

1 fully proved1 ready to prove

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