Golod groups

DefinitionGolod group (Kourovka 12.102)#✓fully proved

Let \(k\) be a field, \(N\) a non-unital associative \(k\)-algebra and \(y_1, \dots, y_r \in N\). The Golod group of \(y\) is the subgroup of the group of units of the unitization \(k \oplus N\) generated by the units that are equal to \(1 + y_i\) for some \(i\). When every \(1 + y_i\) is invertible (for example when \(N\) is nil), this is the subgroup \(\langle 1 + y_1, \dots, 1 + y_r \rangle\) of the adjoint group \(1 + N\).

def golodGroup (k N : Type*) [Field k] [NonUnitalRing N] [Module k N] [IsScalarTower k N N]
    [SMulCommClass k N N] {r : ℕ} (y : Fin r → N) : Subgroup (Unitization k N)ˣ :=
  Subgroup.closure {u | ∃ i, (u : Unitization k N) = 1 + (y i : Unitization k N)}

Used by 1

Used byResidual finiteness of proper quotients of Golod groups (Kourovka 12.102)

DefinitionResidual finiteness of proper quotients of Golod groups (Kourovka 12.102)#✓fully proved

This is the statement that the following holds for every field \(k\) of prime characteristic \(p\), every non-unital associative \(k\)-algebra \(N\), every \(r \ge 2\) and all \(y_1, \dots, y_r \in N\) such that \(N\) is generated by \(y_1, \dots, y_r\) as a non-unital \(k\)-algebra, every element of \(N\) is nilpotent, and for every natural number \(n\) some product of \(n\) elements of \(N\) is nonzero: if \(\Gamma\) is the Golod group of \(y\), then for every nontrivial normal subgroup \(K\) of \(\Gamma\) and every \(x \in \Gamma\) with \(x \notin K\) there is a normal subgroup \(M\) of finite index in \(\Gamma\) with \(K \le M\) and \(x \notin M\). In other words, every proper factor group \(\Gamma / K\) of a Golod group \(\Gamma\) is residually finite. This is Shunkov's question, Kourovka Problem 12.102.

def golodProperQuotientResiduallyFinite : Prop :=
  ∀ (k : Type) [Field k] (p : ℕ) [CharP k p], p.Prime → ∀ (N : Type) [NonUnitalRing N] [Module k N]
    [IsScalarTower k N N] [SMulCommClass k N N] (r : ℕ) (y : Fin r → N), 2 ≤ r →
    NonUnitalAlgebra.adjoin k (Set.range y) = ⊤ →
    (∀ a : N, IsNilpotent (a : Unitization k N)) →
    (∀ n : ℕ, ∃ z : Fin n → N, (List.ofFn fun i => (z i : Unitization k N)).prod ≠ 0) →
    ∀ K : Subgroup (golodGroup k N y), K.Normal → K ≠ ⊥ →
      ∀ x : golodGroup k N y, x ∉ K →
        ∃ M : Subgroup (golodGroup k N y), M.Normal ∧ M.FiniteIndex ∧ K ≤ M ∧ x ∉ M

Uses 1 · Used by 1

Statement usesGolod group (Kourovka 12.102)

Used byNot every proper quotient of a Golod group is residually finite

TheoremNot every proper quotient of a Golod group is residually finite#●ready to prove

The statement that every proper factor group of a Golod group is residually finite is false: there exist a field \(k\) of prime characteristic \(p\), an integer \(r \ge 2\), a non-unital associative \(k\)-algebra \(N\) and elements \(y_1, \dots, y_r \in N\) such that \(N\) is generated by \(y_1, \dots, y_r\) as a non-unital \(k\)-algebra, every element of \(N\) is nilpotent and for every natural number \(n\) some product of \(n\) elements of \(N\) is nonzero, together with a nontrivial normal subgroup \(K\) of the Golod group \(\Gamma\) of \(y\) and an element \(x \in \Gamma\) with \(x \notin K\), such that every normal subgroup of finite index in \(\Gamma\) that contains \(K\) also contains \(x\). This answers Kourovka Problem 12.102 (V. P. Shunkov) in the negative (L. Hammoudi).

public theorem_wanted golodProperQuotientResiduallyFinite_refuted :
    ¬ golodProperQuotientResiduallyFinite

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 12.102 (V. P. Shunkov); resolution: L. Hammoudi, Algebra Colloq. 5 (1998), 371-376, Kourovka Archive, Problem 12.102: "No, not every", https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesResidual finiteness of proper quotients of Golod groups (Kourovka 12.102)

DefinitionGolod group (Kourovka 11.101)#✓fully proved

Let \(k\) be a field, \(N\) a non-unital associative \(k\)-algebra and \(y_1, \dots, y_r \in N\). The Golod group of \(y\) is the subgroup of the group of units of the unitization \(k \oplus N\) generated by the units that are equal to \(1 + y_i\) for some \(i\). When every \(1 + y_i\) is invertible (for example when \(N\) is nil), this is the subgroup \(\langle 1 + y_1, \dots, 1 + y_r \rangle\) of the adjoint group \(1 + N\).

def golodGroup (k N : Type*) [Field k] [NonUnitalRing N] [Module k N] [IsScalarTower k N N]
    [SMulCommClass k N N] {r : ℕ} (y : Fin r → N) : Subgroup (Unitization k N)ˣ :=
  Subgroup.closure {u | ∃ i, (u : Unitization k N) = 1 + (y i : Unitization k N)}

Used by 1

Used byExistence of a Golod group with trivial centre

DefinitionExistence of a Golod group with trivial centre#✓fully proved

Kourovka 11.101 (A. V. Timofeenko), in the strong form proved: some Golod group has trivial centre (the problem asks for finite centre). A Golod group (9.76) is ⟨1 + x₁ + I, …, 1 + x_r + I⟩ ≤ 1 + F/I for r ≥ 2, F the free associative algebra without constant term on x₁, …, x_r over a field of characteristic p > 0 and F/I a non-nilpotent nil-algebra; equivalently N = F/I is a non-unital algebra generated by y₁, …, y_r that is nil but not nilpotent (some product of n elements is nonzero for every n).

def golodGroupWithTrivialCentre : Prop :=
  ∃ (k : Type) (_ : Field k) (p : ℕ) (_ : CharP k p), p.Prime ∧
    ∃ (N : Type) (_ : NonUnitalRing N) (_ : Module k N) (_ : IsScalarTower k N N)
      (_ : SMulCommClass k N N) (r : ℕ) (y : Fin r → N), 2 ≤ r ∧
      NonUnitalAlgebra.adjoin k (Set.range y) = ⊤ ∧
      (∀ a : N, IsNilpotent (a : Unitization k N)) ∧
      (∀ n : ℕ, ∃ z : Fin n → N, (List.ofFn fun i => (z i : Unitization k N)).prod ≠ 0) ∧
      Subgroup.center (golodGroup k N y) = ⊥

Uses 1 · Used by 1

Statement usesGolod group (Kourovka 11.101)

Used byThere is a Golod group with trivial centre

TheoremThere is a Golod group with trivial centre#●ready to prove

There exist a field \(k\) of prime characteristic \(p\), an integer \(r \ge 2\), a non-unital associative \(k\)-algebra \(N\) and elements \(y_1, \dots, y_r \in N\) such that \(N\) is generated by \(y_1, \dots, y_r\) as a non-unital \(k\)-algebra, every element of \(N\) is nilpotent, for every natural number \(n\) some product of \(n\) elements of \(N\) is nonzero, and the Golod group of \(y\) has trivial centre. Since a trivial centre is finite, this answers Kourovka Problem 11.101 (A. V. Timofeenko) positively (V. A. Sereda and A. I. Sozutov).

public theorem_wanted golodGroupWithTrivialCentre_holds : golodGroupWithTrivialCentre

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 11.101 (A. V. Timofeenko); resolution: V. A. Sereda and A. I. Sozutov, Algebra and Logic 45, no. 2 (2006), 134-138, Kourovka Archive, Problem 11.101 ("Yes, there is a Golod group with trivial centre").

Uses 1

Statement usesExistence of a Golod group with trivial centre

4 fully proved2 ready to prove

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