Double commutators of transvections

DefinitionElementary transvection#✓fully proved

For a ring \(R\) and \(n \in \mathbb{N}\), a matrix \(\tau \in \mathrm{GL}_n(R)\) is an elementary transvection when \(\tau = 1 + r e_{ij}\) for some indices \(i \ne j\) and some nonzero \(r \in R\), where \(e_{ij}\) is the matrix unit with \(1\) in position \((i, j)\) and \(0\) elsewhere.

def IsElementaryTransvection {R : Type*} [Ring R] {n : ℕ} (τ : GL (Fin n) R) : Prop :=
  ∃ (i j : Fin n) (r : R), i ≠ j ∧ r ≠ 0 ∧ (τ : Matrix (Fin n) (Fin n) R) = 1 + Matrix.single i j r

Used by 1

Used byPetechuk's double commutator question (Kourovka 10.46)

DefinitionPetechuk's double commutator question (Kourovka 10.46)#✓fully proved

The statement that for every nontrivial associative ring \(R\) with identity, every \(n \ge 3\) and every \(\sigma \in \mathrm{GL}_n(R)\) there are elementary transvections \(\tau_1, \tau_2 \in \mathrm{GL}_n(R)\) such that \([[\sigma, \tau_1], \tau_2]\) is unipotent, that is, \([[\sigma, \tau_1], \tau_2] - 1\) is a nilpotent matrix, where \([x, y] = x y x^{-1} y^{-1}\).

In the problem, transvections are read as elementary transvections.

def doubleCommutatorWithTransvectionsUnipotent : Prop :=
  ∀ (R : Type) [Ring R] [Nontrivial R] (n : ℕ), 3 ≤ n → ∀ σ : GL (Fin n) R,
    ∃ τ₁ τ₂ : GL (Fin n) R, IsElementaryTransvection τ₁ ∧ IsElementaryTransvection τ₂ ∧
      IsNilpotent (((σ * τ₁ * σ⁻¹ * τ₁⁻¹) * τ₂ * (σ * τ₁ * σ⁻¹ * τ₁⁻¹)⁻¹ * τ₂⁻¹ : GL (Fin n) R) -
        1 : Matrix (Fin n) (Fin n) R)

Uses 1 · Used by 1

Statement usesElementary transvection

Used byPetechuk's double commutator question has a negative answer (Kourovka 10.46)

TheoremPetechuk's double commutator question has a negative answer (Kourovka 10.46)#●ready to prove

There are a nontrivial associative ring \(R\) with identity, an integer \(n \ge 3\) and a matrix \(\sigma \in \mathrm{GL}_n(R)\) such that for all elementary transvections \(\tau_1, \tau_2 \in \mathrm{GL}_n(R)\) the matrix \([[\sigma, \tau_1], \tau_2] - 1\) is not nilpotent, that is, \([[\sigma, \tau_1], \tau_2]\) is not unipotent. Here \([x, y] = x y x^{-1} y^{-1}\), and an elementary transvection is a matrix \(1 + r e_{ij}\) with \(i \ne j\) and \(r \in R\) nonzero, where \(e_{ij}\) is the matrix unit with \(1\) in position \((i, j)\) and \(0\) elsewhere.

Muliarchyk (Theorem 1.1) shows the following for every field \(k\) and \(n \ge 2\). Let \(R\) be the free associative \(k\)-algebra on generators \(x_{ij}, y_{ij}\) (\(1 \le i, j \le n\)) modulo the relations \(XY = YX = I\), where \(X = (x_{ij})\), \(Y = (y_{ij})\) and \(I\) is the \(n \times n\) identity matrix; \(R\) is nonzero. Then every nonempty iterated commutator of \(X \in \mathrm{GL}_n(R)\) with nonidentity unipotent matrices is nonunipotent.

public theorem_wanted doubleCommutatorWithTransvectionsUnipotent_refuted :
    ¬ doubleCommutatorWithTransvectionsUnipotent

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 10.46 (V. M. Petechuk); resolution: K. Muliarchyk, A counterexample to Kourovka Notebook Problem 10.46: unipotent commutators over noncommutative algebras, preprint (2026), Theorem 1.1 (over R = k⟨x_ij, y_ij⟩ / (XY = YX = I) every nonempty iterated commutator of X with nonidentity unipotent matrices is nonunipotent); the Kourovka Notebook (21st issue) marks 10.46 solved, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka_10.46_solution.pdf.

Uses 1

Statement usesPetechuk's double commutator question (Kourovka 10.46)

2 fully proved1 ready to prove

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