Coset-free partitions of groups

DefinitionKourovka Problem 13.46: coset-free partitions#✓fully proved

The proposition that every uncountable abelian group \(G\) of finite odd exponent can be written as \(G = A_1 \cup A_2\) with \(A_1\) and \(A_2\) disjoint, so that for every infinite subgroup \(H\) of \(G\) and every \(g \in G\), the coset \(gH\) is contained neither in \(A_1\) nor in \(A_2\). The problem is due to I. V. Protasov.

def oddExponentCosetFreePartition : Prop :=
  ∀ (G : Type) [CommGroup G], Uncountable G → Monoid.exponent G ≠ 0 →
    Odd (Monoid.exponent G) →
    ∃ A₁ A₂ : Set G, A₁ ∪ A₂ = Set.univ ∧ Disjoint A₁ A₂ ∧
      ∀ H : Subgroup G, (H : Set G).Infinite → ∀ g : G,
        ¬ g • (H : Set G) ⊆ A₁ ∧ ¬ g • (H : Set G) ⊆ A₂

Used by 1

Used byUncountable abelian groups of odd exponent have coset-free partitions

TheoremUncountable abelian groups of odd exponent have coset-free partitions#●ready to prove

Every uncountable abelian group \(G\) of finite odd exponent can be written as \(G = A_1 \cup A_2\) with \(A_1\) and \(A_2\) disjoint, so that for every infinite subgroup \(H\) of \(G\) and every \(g \in G\), the coset \(gH\) is contained neither in \(A_1\) nor in \(A_2\). This answers Kourovka Problem 13.46 in the affirmative.

public theorem_wanted oddExponentCosetFreePartition_holds : oddExponentCosetFreePartition

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 13.46 (I. V. Protasov); resolution: E. G. Zelenyuk, Math. Notes 67 (2000), 599-602, Kourovka Archive, Problem 13.46 (answer: Yes, it can).

Uses 1

Statement usesKourovka Problem 13.46: coset-free partitions

1 fully proved1 ready to prove

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