Group topologies

DefinitionKourovka Problem 15.79 (Protasov's question on the integers)#✓fully proved

The statement that there is a topology on \(\mathbb{Z}\) that makes \((\mathbb{Z}, +)\) a Hausdorff topological group and in which the sequence \((2^n + 3^n)_{n \in \mathbb{N}}\) converges to \(0\).

def integerGroupTopologyPowSumTendstoZero : Prop :=
  ∃ t : TopologicalSpace ℤ, letI := t; IsTopologicalAddGroup ℤ ∧ T2Space ℤ ∧
    Filter.Tendsto (fun n : ℕ => (2 : ℤ) ^ n + 3 ^ n) Filter.atTop (nhds 0)
Used by 1
Used byA Hausdorff group topology on the integers in which 2ⁿ + 3ⁿ tends to zero

TheoremA Hausdorff group topology on the integers in which 2ⁿ + 3ⁿ tends to zero#●ready to prove

There is a topology on \(\mathbb{Z}\) that makes \((\mathbb{Z}, +)\) a Hausdorff topological group and in which the sequence \((2^n + 3^n)_{n \in \mathbb{N}}\) converges to \(0\). That is, Kourovka Problem 15.79 has a positive answer.

The Kourovka Archive answers 15.79 "Yes, there does", as follows from Theorem 3 of Ruzsa; another solution is due to Skresanov.

public theorem_wanted integerGroupTopologyPowSumTendstoZero_holds :
    integerGroupTopologyPowSumTendstoZero
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 15.79 (I. V. Protasov); resolution: I. Z. Ruzsa, in: Proc. Conf. Number Theory (Budapest, 1987), Vol. I, Colloq. Math. Soc. János Bolyai 51, North-Holland, Amsterdam, 1990, 473-504; S. V. Skresanov, Group topologies on integers and S-unit equations, Siberian Math. J. 61 (2020), no. 3, 542-544, arXiv:1911.11963, Ruzsa, Theorem 3, as applied in the Kourovka Archive, Problem 15.79 (answer: Yes, there does); another solution by Skresanov, abstract (for a finite set of primes S, a Hausdorff group topology on ℤ in which every growing sequence of S-integers tends to 0; as a corollary, 2ⁿ + 3ⁿ tends to 0 in a Hausdorff group topology), https://arxiv.org/abs/1911.11963.
Uses 1
Statement usesKourovka Problem 15.79 (Protasov's question on the integers)

DefinitionKourovka Problem 14.4(a) (nilpotent groups with non-modular topology lattice)#✓fully proved

This proposition states that some nilpotent group \(G\) has a non-modular lattice \(\mathcal{L}(G)\) of all (not necessarily Hausdorff) group topologies.

def nilpotentGroupTopologyLatticeNotModular : Prop :=
  ∃ (G : Type) (_ : Group G), Group.IsNilpotent G ∧ ¬ IsModularLattice (GroupTopology G)
Used by 1
Used byA nilpotent group with non-modular lattice of group topologies

DefinitionKourovka Problem 14.4(b) (countable nilpotent groups)#✓fully proved

This proposition states that for every countably infinite non-abelian nilpotent group \(G\), the lattice \(\mathcal{L}(G)\) of all (not necessarily Hausdorff) group topologies on \(G\) is not modular.

def countableNilpotentNonabelianTopologyLatticeNotModular : Prop :=
  ∀ (G : Type) [Group G], Countable G → Infinite G → Group.IsNilpotent G →
    (∃ x y : G, x * y ≠ y * x) → ¬ IsModularLattice (GroupTopology G)
Used by 1
Used byA countable nilpotent group with modular lattice of group topologies

TheoremA nilpotent group with non-modular lattice of group topologies#●ready to prove

There is a nilpotent group \(G\) whose lattice of all (not necessarily Hausdorff) group topologies is not modular.

Arnautov and Topală found such a group; the Kourovka Notebook (21st issue) answers 14.4(a) "Yes".

public theorem_wanted nilpotentGroupTopologyLatticeNotModular_holds :
    nilpotentGroupTopologyLatticeNotModular
Sources
  • E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.4 (V. I. Arnautov); resolution: (a) V. Arnautov and A. Topală, Bul. Acad. Ştiinţe Repub. Mold. Mat. 1998, no. 2(27), 130-131; (b) D. Peng, On the lattice of group topologies, preprint, arXiv:2310.08269 (2023), (b) Theorem 1.1: there is a countably infinite non-abelian nilpotent group whose lattice of all (not necessarily Hausdorff) group topologies is modular; the Kourovka Notebook (21st issue) marks 14.4 solved: a) Yes; b) No, there are such groups with modular L(G), https://arxiv.org/abs/2310.08269.
Uses 1
Statement usesKourovka Problem 14.4(a) (nilpotent groups with non-modular topology lattice)

TheoremA countable nilpotent group with modular lattice of group topologies#●ready to prove

It is not the case that for every countably infinite non-abelian nilpotent group \(G\) the lattice of all (not necessarily Hausdorff) group topologies on \(G\) is not modular; that is, some countably infinite non-abelian nilpotent group has a modular lattice of group topologies.

Peng constructs (Theorem 1.1) a countably infinite non-abelian nilpotent group whose lattice of group topologies is modular; the Kourovka Notebook (21st issue) answers 14.4(b) "No".

public theorem_wanted countableNilpotentNonabelianTopologyLatticeNotModular_refuted :
    ¬ countableNilpotentNonabelianTopologyLatticeNotModular
Sources
  • E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.4 (V. I. Arnautov); resolution: (a) V. Arnautov and A. Topală, Bul. Acad. Ştiinţe Repub. Mold. Mat. 1998, no. 2(27), 130-131; (b) D. Peng, On the lattice of group topologies, preprint, arXiv:2310.08269 (2023), (b) Theorem 1.1: there is a countably infinite non-abelian nilpotent group whose lattice of all (not necessarily Hausdorff) group topologies is modular; the Kourovka Notebook (21st issue) marks 14.4 solved: a) Yes; b) No, there are such groups with modular L(G), https://arxiv.org/abs/2310.08269.
Uses 1
Statement usesKourovka Problem 14.4(b) (countable nilpotent groups)
3 fully proved3 ready to prove

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