Groups of exponent 4

DefinitionExponent 4 detected on boundedly generated subgroups (Kourovka 13.10)#✓fully proved

The proposition that there is a function \(f : \mathbb{N} \to \mathbb{N}\) with the following property: for every \(k \in \mathbb{N}\), every group \(G\) whose \(k\)-th derived subgroup \(G^{(k)}\) is trivial (a soluble group of derived length at most \(k\)), and every generating set \(A\) of \(G\), if \(x^4 = 1\) for every element \(x\) of every subgroup generated by a finite subset of \(A\) with at most \(f(k)\) elements, then \(x^4 = 1\) for every \(x \in G\). This is Kourovka Notebook Problem 13.10 (V. V. Bludov).

def exponentFourDetectedOnBoundedSubsets : Prop :=
  ∃ f : ℕ → ℕ, ∀ (k : ℕ) (G : Type) [Group G], derivedSeries G k = ⊥ →
    ∀ A : Set G, Subgroup.closure A = ⊤ →
      (∀ S : Finset G, (S : Set G) ⊆ A → S.card ≤ f k →
        ∀ g ∈ Subgroup.closure (S : Set G), g ^ 4 = 1) →
      ∀ g : G, g ^ 4 = 1

Used by 1

Used byExponent 4 is not detected on boundedly generated subgroups

TheoremExponent 4 is not detected on boundedly generated subgroups#●ready to prove

There is no function \(f : \mathbb{N} \to \mathbb{N}\) with the following property: for every \(k \in \mathbb{N}\), every group \(G\) whose \(k\)-th derived subgroup \(G^{(k)}\) is trivial, and every generating set \(A\) of \(G\), if \(x^4 = 1\) for every element \(x\) of every subgroup generated by a finite subset of \(A\) with at most \(f(k)\) elements, then \(x^4 = 1\) for every \(x \in G\). This is the negative answer to Kourovka Notebook Problem 13.10 (V. V. Bludov), given by G. S. Deryabina and A. N. Krasil'nikov (2003).

public theorem_wanted exponentFourDetectedOnBoundedSubsets_refuted :
    ¬ exponentFourDetectedOnBoundedSubsets

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 13.10 (V. V. Bludov); resolution: G. S. Deryabina and A. N. Krasil'nikov, Siberian Math. J. 44 (2003), no. 1, 58-60, Kourovka Archive, Problem 13.10: No, there is not.

Uses 1

Statement usesExponent 4 detected on boundedly generated subgroups (Kourovka 13.10)

1 fully proved1 ready to prove

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