Dimension subgroups

DefinitionAugmentation ideal of the integral group ring#✓fully proved

For a group \(G\), the augmentation ideal \(\Delta(G)\) of the integral group ring \(\mathbb{Z}G\) is the \(\mathbb{Z}\)-submodule spanned by the elements \(g - 1\) for \(g \in G\). It is the kernel of the augmentation map \(\mathbb{Z}G \to \mathbb{Z}\), \(\sum_g a_g g \mapsto \sum_g a_g\).

noncomputable def augmentationIdeal (G : Type*) [Group G] : Submodule ℤ (MonoidAlgebra ℤ G) :=
  Submodule.span ℤ (Set.range fun g : G => MonoidAlgebra.of ℤ G g - 1)

Used by 1

Used byDimension subgroups

DefinitionDimension subgroups#✓fully proved

Let \(\Delta(G)\) be the augmentation ideal of the integral group ring \(\mathbb{Z}G\) of a group \(G\), and for \(n \in \mathbb{N}\) let \(\Delta(G)^n\) be its \(n\)-th power, the \(\mathbb{Z}\)-span of all products of \(n\) elements of \(\Delta(G)\). An element \(g \in G\) lies in the \(n\)-th dimension subgroup \(D_n(G) = G \cap (1 + \Delta(G)^n)\) when \(g - 1 \in \Delta(G)^n\).

def MemDimensionSubgroup {G : Type*} [Group G] (n : ℕ) (g : G) : Prop :=
  MonoidAlgebra.of ℤ G g - 1 ∈ augmentationIdeal G ^ n

Uses 1 · Used by 2

Statement usesAugmentation ideal of the integral group ring

Used byGupta–Kuz'min centrality question (Kourovka 12.22(a)) · Gupta–Kuz'min exponent question (Kourovka 12.22(b))

DefinitionGupta–Kuz'min centrality question (Kourovka 12.22(a))#✓fully proved

The statement that for every group \(G\), every \(n \ge 1\), every element \(x\) of the \(n\)-th dimension subgroup \(D_n(G)\) and every \(g \in G\), the commutator \(x g x^{-1} g^{-1}\) lies in \(\gamma_n(G)\), the \(n\)-th term of the lower central series of \(G\), indexed so that \(\gamma_1(G) = G\) and \(\gamma_{k+1}(G) = [\gamma_k(G), G]\). Equivalently, the image of \(D_n(G)\) in \(G/\gamma_n(G)\) is central.

def dimensionQuotientCentral : Prop :=
  ∀ (G : Type) [Group G] (n : ℕ), 1 ≤ n → ∀ x : G, MemDimensionSubgroup n x →
    ∀ g : G, x * g * x⁻¹ * g⁻¹ ∈ (⊤ : Subgroup G).lowerCentralSeries (n - 1)

Uses 1 · Used by 1

Statement usesDimension subgroups

Used byDimension quotients need not be central

DefinitionGupta–Kuz'min exponent question (Kourovka 12.22(b))#✓fully proved

The statement that for every group \(G\), every \(n \ge 1\) and every element \(x\) of the \(n\)-th dimension subgroup \(D_n(G)\), the square \(x^2\) lies in \(\gamma_n(G)\), the \(n\)-th term of the lower central series of \(G\), indexed so that \(\gamma_1(G) = G\) and \(\gamma_{k+1}(G) = [\gamma_k(G), G]\). Equivalently, the image of \(D_n(G)\) in \(G/\gamma_n(G)\) has exponent dividing \(2\).

def dimensionQuotientExponentTwo : Prop :=
  ∀ (G : Type) [Group G] (n : ℕ), 1 ≤ n → ∀ x : G, MemDimensionSubgroup n x →
    x ^ 2 ∈ (⊤ : Subgroup G).lowerCentralSeries (n - 1)

Uses 1 · Used by 1

Statement usesDimension subgroups

Used byDimension quotients need not have exponent dividing 2

TheoremDimension quotients need not be central#●ready to prove

There exist a group \(G\), an integer \(n \ge 1\), an element \(x\) of the \(n\)-th dimension subgroup \(D_n(G)\) and an element \(g \in G\) such that the commutator \(x g x^{-1} g^{-1}\) does not lie in \(\gamma_n(G)\), the \(n\)-th term of the lower central series of \(G\) (indexed so that \(\gamma_1(G) = G\)). That is, the answer to Kourovka Problem 12.22(a) is negative: the image of \(D_n(G)\) in \(G/\gamma_n(G)\) need not be central.

public theorem_wanted dimensionQuotientCentral_refuted : ¬ dimensionQuotientCentral

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 12.22 (N. D. Gupta, Yu. V. Kuz'min); resolution: N. D. Gupta and Yu. V. Kuz'min, J. Pure Appl. Algebra 104, no. 1 (1995), 191-197 (part a); L. Bartholdi and R. Mikhailov, J. Topology 16, no. 2 (2023), 822-853 (part b), Kourovka Archive, Problem 12.22: "a) No, not always; moreover, D_n(G)/γ_n(G) need not be contained in any term of the upper central series of G/γ_n(G) with fixed number" and "b) No, not always", https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesGupta–Kuz'min centrality question (Kourovka 12.22(a))

TheoremDimension quotients need not have exponent dividing 2#●ready to prove

There exist a group \(G\), an integer \(n \ge 1\) and an element \(x\) of the \(n\)-th dimension subgroup \(D_n(G)\) such that \(x^2\) does not lie in \(\gamma_n(G)\), the \(n\)-th term of the lower central series of \(G\) (indexed so that \(\gamma_1(G) = G\)). That is, the answer to Kourovka Problem 12.22(b) is negative: the image of \(D_n(G)\) in \(G/\gamma_n(G)\) need not have exponent dividing \(2\).

public theorem_wanted dimensionQuotientExponentTwo_refuted : ¬ dimensionQuotientExponentTwo

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 12.22 (N. D. Gupta, Yu. V. Kuz'min); resolution: N. D. Gupta and Yu. V. Kuz'min, J. Pure Appl. Algebra 104, no. 1 (1995), 191-197 (part a); L. Bartholdi and R. Mikhailov, J. Topology 16, no. 2 (2023), 822-853 (part b), Kourovka Archive, Problem 12.22: "a) No, not always; moreover, D_n(G)/γ_n(G) need not be contained in any term of the upper central series of G/γ_n(G) with fixed number" and "b) No, not always", https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesGupta–Kuz'min exponent question (Kourovka 12.22(b))

4 fully proved2 ready to prove

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