Equally distributed elements of free metabelian groups

AbbreviationFree metabelian group of rank \(n\) (Kourovka 19.94)#✓fully proved

For a natural number \(n\), let \(F_n\) be the free group on \(n\) generators. The free metabelian group of rank \(n\) is the quotient \(M_n = F_n / F_n''\) of \(F_n\) by its second derived subgroup \(F_n'' = [[F_n, F_n], [F_n, F_n]]\).

abbrev FreeMetabelian (n : ℕ) :=
  FreeGroup (Fin n) ⧸ derivedSeries (FreeGroup (Fin n)) 2

Used by 1

Used byTimoshenko's question on equally distributed elements (Kourovka 19.94)

DefinitionWords with the same distribution#✓fully proved

Let \(n\) be a natural number and \(F_n\) the free group on generators \(x_1, \dots, x_n\). Words \(w_1, w_2 \in F_n\) have the same distribution when, for every finite metabelian group \(G\) (that is, \(G'' = 1\)) and every \(g \in G\), the number of tuples \((a_1, \dots, a_n) \in G^n\) with \(w_1(a_1, \dots, a_n) = g\) equals the number of tuples with \(w_2(a_1, \dots, a_n) = g\). Here \(w(a_1, \dots, a_n)\) is the image of \(w\) under the homomorphism \(F_n \to G\) sending \(x_i\) to \(a_i\).

Since \(G\) is metabelian, this depends only on the images of \(w_1, w_2\) in the free metabelian group \(F_n / F_n''\).

def SameDistribution {n : ℕ} (w₁ w₂ : FreeGroup (Fin n)) : Prop :=
  ∀ (G : Type) [Group G] [Finite G], derivedSeries G 2 = ⊥ → ∀ g : G,
    Nat.card {a : Fin n → G // FreeGroup.lift a w₁ = g} =
      Nat.card {a : Fin n → G // FreeGroup.lift a w₂ = g}

Used by 1

Used byTimoshenko's question on equally distributed elements (Kourovka 19.94)

DefinitionTimoshenko's question on equally distributed elements (Kourovka 19.94)#✓fully proved

The statement that for every natural number \(n\) and all words \(w_1, w_2\) in the free group \(F_n\) of rank \(n\), the words \(w_1\) and \(w_2\) have the same distribution if and only if some automorphism of the free metabelian group \(M_n = F_n / F_n''\) maps the image of \(w_1\) in \(M_n\) to the image of \(w_2\).

def sameDistributionIffAutomorphic : Prop :=
  ∀ (n : ℕ) (w₁ w₂ : FreeGroup (Fin n)), SameDistribution w₁ w₂ ↔
    ∃ φ : MulAut (FreeMetabelian n), φ (QuotientGroup.mk w₁) = QuotientGroup.mk w₂

Uses 2 · Used by 1

Statement usesFree metabelian group of rank \(n\) (Kourovka 19.94) · Words with the same distribution

Used byEqually distributed elements need not be automorphic (Kourovka 19.94)

TheoremEqually distributed elements need not be automorphic (Kourovka 19.94)#●ready to prove

It is not the case that for every natural number \(n\) and all words \(w_1, w_2\) in the free group \(F_n\) of rank \(n\), the words \(w_1\) and \(w_2\) have the same distribution if and only if some automorphism of the free metabelian group \(M_n = F_n / F_n''\) maps the image of \(w_1\) in \(M_n\) to the image of \(w_2\).

Ionin and Semidetnov (Answer 2.2) give two elements of the free metabelian group of rank \(2\) with the same distribution on every finite metabelian group that no automorphism maps one to the other (the converse direction is automatic). The solution was found with the assistance of large language models and checked by the authors; the 21st issue of the Kourovka Notebook marks 19.94 solved ("No, it is not true").

public theorem_wanted sameDistributionIffAutomorphic_refuted : ¬ sameDistributionIffAutomorphic

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 19.94 (E. I. Timoshenko); resolution: V. Ionin and A. Semidetnov, On Some More Problems from the Kourovka Notebook, preprint, arXiv:2608.29219 (2026), Section 2, Answer 2.2 with Proposition 2.3 and Lemmas 2.6-2.7: in M₂ = ⟨x, y⟩ with c = [x, y] and a^z = z⁻¹az, the elements g₁ = c^{x²y⁷}(c^x)⁻¹(c^{xy⁷})⁻¹c and g₂ = c^{x³y⁷}(c^x)⁻¹(c^{x²y⁷})⁻¹c have the same distribution on every finite metabelian group but lie in distinct Aut(M₂)-orbits; found with the assistance of large language models and checked by the authors; the Kourovka Notebook (21st issue) marks 19.94 solved: No, it is not true, https://arxiv.org/abs/2608.29219.

Uses 1

Statement usesTimoshenko's question on equally distributed elements (Kourovka 19.94)

3 fully proved1 ready to prove

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