FormalConjectures.Kourovka.«1

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import FormalConjecturesUtil

Conjecture 1.35(c)

by A. I. Mal'cev and L. Fuchs

Reference: The Kourovka Notebook

namespace Kourovka.«1.35c»

A relation on a group is bi-invariant if it is preserved by both left and right multiplication.

def IsBiInvariant (G : Type*) [Group G] (r : G → G → Prop) : Prop :=
  (∀ g a b, r a b → r (g * a) (g * b)) ∧
  (∀ g a b, r a b → r (a * g) (b * g))

ProOrderable G means every bi-invariant partial order extends to a bi-invariant linear order.

def ProOrderable (G : Type*) [Group G] : Prop :=
  ∀ r : G → G → Prop,
    IsPartialOrder G r →
    IsBiInvariant G r →
    ∃ s : G → G → Prop,
      IsLinearOrder G s ∧ IsBiInvariant G s ∧ ∀ x y, r x y → s x y

Do there exist simple pro-orderable groups?

@[category research open, AMS 20]
theorem kourovka_1_35c : answer(sorry) ↔
    ∃ (G : Type) (_ : Group G), IsSimpleGroup G ∧ ProOrderable G := by⊢ True ↔ ∃ G x, IsSimpleGroup G ∧ ProOrderable G
  sorryAll goals completed! 🐙

end Kourovka.«1.35c»