Unique product groups

DefinitionLeft-orderable groups#✓fully proved

A group \(G\) is left-orderable if it admits a linear order \(\le\) such that \(a \le b\) implies \(ca \le cb\) for all \(a, b, c \in G\).

def IsLeftOrderable (G : Type*) [Group G] : Prop :=
  ∃ r : LinearOrder G, letI := r; ∀ a b c : G, a ≤ b → c * a ≤ c * b

Used by 1

Used byKourovka Problem 15.49 (unique product groups and left orders)

DefinitionKourovka Problem 15.49 (unique product groups and left orders)#✓fully proved

The statement, asked by P. Linnell, that there exists a group \(G\) with unique products that is not left-orderable. Here \(G\) has unique products if for all finite nonempty subsets \(X, Y \subseteq G\) there are \(x_0 \in X\) and \(y_0 \in Y\) such that \((x_0, y_0)\) is the only pair \((x, y) \in X \times Y\) with \(xy = x_0 y_0\).

def uniqueProductGroupNotLeftOrderable : Prop :=
  ∃ (G : Type) (_ : Group G), UniqueProds G ∧ ¬ IsLeftOrderable G

Uses 1 · Used by 1

Statement usesLeft-orderable groups

Used byA unique product group need not be left-orderable

TheoremA unique product group need not be left-orderable#●ready to prove

There exists a group \(G\) that has unique products but is not left-orderable: for all finite nonempty subsets \(X, Y \subseteq G\) there are \(x_0 \in X\) and \(y_0 \in Y\) such that \((x_0, y_0)\) is the only pair \((x, y) \in X \times Y\) with \(xy = x_0 y_0\), and yet \(G\) admits no linear order invariant under left multiplication.

public theorem_wanted uniqueProductGroupNotLeftOrderable_holds : uniqueProductGroupNotLeftOrderable

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 15.49 (P. Linnell); resolution: N. Dunfield, Appendix B in S. Kionke and J. Raimbault, On geometric aspects of diffuse groups, Doc. Math. 21 (2016), 873-915, arXiv:1411.6449, Section 2.2 and the appendix (diffuse groups have unique products, by Bowditch's observation; the appendix gives a compact hyperbolic 3-manifold group that is diffuse but not left-orderable); Kourovka Archive, Problem 15.49 (answer: Yes, there does), https://arxiv.org/abs/1411.6449.

Uses 1

Statement usesKourovka Problem 15.49 (unique product groups and left orders)

2 fully proved1 ready to prove

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