Partitions meeting every infinite coset

DefinitionKourovka 16.43: partitioning \(\bigoplus_{\omega_1} \mathbb{Z}/3\mathbb{Z}\) into three parts meeting every infinite coset#✓fully proved

Kourovka 16.43: partitioning \(\bigoplus_{\omega_1} \mathbb{Z}/3\mathbb{Z}\) into three parts meeting every infinite coset. The proposition that for every set \(I\) of cardinality \(\aleph_1\), the group \(A = \bigoplus_{i \in I} \mathbb{Z}/3\mathbb{Z}\) of finitely supported functions from \(I\) to \(\mathbb{Z}/3\mathbb{Z}\) can be partitioned into three subsets \(A_0, A_1, A_2\) such that for every infinite subgroup \(H\) of \(A\) and every \(x \in A\), the coset \(x + H\) meets each \(A_i\).

def threePartitionMeetsInfiniteCosets : Prop :=
  ∀ ι : Type, Cardinal.mk ι = Cardinal.aleph 1 →
    ∃ c : (ι →₀ ZMod 3) → Fin 3, ∀ H : AddSubgroup (ι →₀ ZMod 3),
      (H : Set (ι →₀ ZMod 3)).Infinite → ∀ (x : ι →₀ ZMod 3) (i : Fin 3), ∃ h ∈ H, c (x + h) = i

Used by 1

Used byA partition of \(\bigoplus_{\omega_1} \mathbb{Z}/3\mathbb{Z}\) into three parts meeting every infinite coset

TheoremA partition of \(\bigoplus_{\omega_1} \mathbb{Z}/3\mathbb{Z}\) into three parts meeting every infinite coset#●ready to prove

A partition of \(\bigoplus_{\omega_1} \mathbb{Z}/3\mathbb{Z}\) into three parts meeting every infinite coset. Let \(I\) be a set of cardinality \(\aleph_1\) and let \(A = \bigoplus_{i \in I} \mathbb{Z}/3\mathbb{Z}\) be the group of finitely supported functions from \(I\) to \(\mathbb{Z}/3\mathbb{Z}\). Then \(A\) can be partitioned into three subsets \(A_0, A_1, A_2\) such that for every infinite subgroup \(H\) of \(A\) and every \(x \in A\), the coset \(x + H\) meets each \(A_i\).

The Kourovka Archive records more: every infinite abelian group with finitely many involutions can be partitioned into infinitely many subsets such that every coset modulo an infinite subgroup meets each subset of the partition.

public theorem_wanted threePartitionMeetsInfiniteCosets_holds : threePartitionMeetsInfiniteCosets

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 16.43 (E. G. Zelenyuk); resolution: Y. Zelenyuk, J. Combin. Theory Ser. A 115 (2008), 331-339, Kourovka Archive, Problem 16.43 (Yes; moreover every infinite abelian group with finitely many involutions can be partitioned into infinitely many subsets such that every coset modulo an infinite subgroup meets each subset).

Uses 1

Statement usesKourovka 16.43: partitioning \(\bigoplus_{\omega_1} \mathbb{Z}/3\mathbb{Z}\) into three parts meeting every infinite coset

1 fully proved1 ready to prove

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