FormalConjectures.Kourovka.«1

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import FormalConjecturesUtil

Conjecture 1.74 (Tarski monster topologizability)

by V. P. Platonov

Problem 1.74 asks to describe all "minimal topological groups" in Platonov's sense: non-discrete Hausdorff topological groups all of whose proper closed subgroups are discrete. A natural test case: does there exist a Tarski monster group admitting a non-discrete Hausdorff group topology? A Tarski monster would be a minimal group in this sense, since all its proper subgroups are finite (hence discrete in any Hausdorff group topology).

Reference: The Kourovka Notebook

namespace Kourovka.«1.74»

A Tarski monster group: an infinite group in which every non-trivial proper subgroup has order a fixed prime $p$.

def IsTarskiMonster (G : Type*) [Group G] : Prop :=
  Infinite G ∧ ∃ p : ℕ, p.Prime ∧
    ∀ H : Subgroup G, H ≠ ⊥ → H ≠ ⊤ → Nat.card H = p

Does there exist a Tarski monster group that admits a non-discrete Hausdorff group topology?

@[category research open, AMS 20 22]
theorem kourovka_1_74 : answer(sorry) ↔
    ∃ (G : Type) (_ : Group G) (_ : TopologicalSpace G),
      IsTarskiMonster G ∧ IsTopologicalGroup G ∧ T2Space G ∧
      ¬ DiscreteTopology G := by⊢ True ↔ ∃ G x x_1, IsTarskiMonster G ∧ IsTopologicalGroup G ∧ T2Space G ∧ ¬DiscreteTopology G
  sorryAll goals completed! 🐙

end Kourovka.«1.74»