Direct products of subsets of groups

DefinitionDirect product of subsets#✓fully proved

For subsets \(A\) and \(B\) of a group \(G\), the product \(AB = \{ab \mid a \in A,\ b \in B\}\) is direct when each element of \(AB\) is \(ab\) for unique \(a \in A\) and \(b \in B\): whenever \(a_1 b_1 = a_2 b_2\) with \(a_1, a_2 \in A\) and \(b_1, b_2 \in B\), we have \(a_1 = a_2\) and \(b_1 = b_2\).

def IsDirectProduct {G : Type} [Group G] (A B : Set G) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ b₁ ∈ B, ∀ b₂ ∈ B, a₁ * b₁ = a₂ * b₂ → a₁ = a₂ ∧ b₁ = b₂

Used by 1

Used byInfinite groups with equicardinal maximal direct factors (Kourovka 21.58)

DefinitionInfinite groups with equicardinal maximal direct factors (Kourovka 21.58)#✓fully proved

The statement that there is an infinite group \(G\) such that for every subset \(A \subseteq G\), any two subsets \(B_1, B_2 \subseteq G\) that are maximal under inclusion among the subsets \(B\) for which the product \(AB\) is direct have the same cardinality.

def existsInfiniteGroupStableFactors : Prop :=
  ∃ (G : Type) (_ : Group G), Infinite G ∧ ∀ A B₁ B₂ : Set G,
    Maximal (IsDirectProduct A) B₁ → Maximal (IsDirectProduct A) B₂ →
      Cardinal.mk B₁ = Cardinal.mk B₂

Uses 1 · Used by 1

Statement usesDirect product of subsets

Used byNo infinite group has equicardinal maximal direct factors

TheoremNo infinite group has equicardinal maximal direct factors#●ready to prove

For every infinite group \(G\) there are subsets \(A, B_1, B_2 \subseteq G\) such that \(B_1\) and \(B_2\) are both maximal under inclusion among the subsets \(B\) for which the product \(AB\) is direct, but \(B_1\) and \(B_2\) have different cardinalities. That is, the answer to Kourovka Problem 21.58 is negative.

public theorem_wanted existsInfiniteGroupStableFactors_refuted : ¬ existsInfiniteGroupStableFactors

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 21.58 (M. H. Hooshmand); resolution: M. I. Kabenyuk, Factors in infinite groups, preprint, arXiv:2602.22876 (2026); M. H. Hooshmand, preprint, arXiv:2604.08724 (2026), Remark 1.16, Kabenyuk, Section 6, Theorem: every infinite group is unstable, i.e. has a subset A whose maximal subsets U with AU direct do not all have the same cardinality; Kourovka Notebook (21st issue), Problem 21.58: No, there are no such groups, https://arxiv.org/abs/2602.22876.

Uses 1

Statement usesInfinite groups with equicardinal maximal direct factors (Kourovka 21.58)

2 fully proved1 ready to prove

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