1_74.lean`

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4 declarations, 2 contain sorry, 0 declared as axioms

Conjecture 1.74 (Minimal topological groups)

by V. P. Platonov

Describe all "minimal topological groups", that is, non-discrete Hausdorff topological groups all of whose proper closed subgroups are discrete. The minimal locally compact groups can be described without much effort, but the problem is probably complicated in the general case.

A Tarski monster group equipped with a non-discrete Hausdorff group topology is a minimal topological group in this sense, since all its proper subgroups are finite (hence discrete in any Hausdorff group topology). Such topologizable Tarski monsters exist by a theorem of Klyachko, Olshanskii and Osin.

References:

The Kourovka Notebook
A. A. Klyachko, A. Yu. Olshanskii, D. V. Osin, On topologizable and non-topologizable groups, Topology Appl. 160 (2013), 2104–2120, arXiv:1210.7895, Theorem 1.4.

Imports (1)

FormalConjecturesUtil

def Kourovka.«1.74».IsMinimalTopologicalGroup A minimal topological group in Platonov's sense: a non-discrete Hausdorff topological group all of whose proper closed subgroups are discrete line 50 source
Source documentation
A minimal topological group in Platonov's sense: a non-discrete Hausdorff topological group all of whose proper closed subgroups are discrete.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace
Proof uses: And, IsTopologicalGroup, T2Space, Not, DiscreteTopology, Subgroup, Ne, Top.top, IsClosed, Subtype, Membership.mem
5 instances, coercions and auxiliary constants left out.
```
def IsMinimalTopologicalGroup (G : Type*) [Group G] [TopologicalSpace G] : Prop :=
…
```
theorem Kourovka.«1.74».kourovka_1_74 Describe all minimal topological groups, that is, all non-discrete Hausdorff topological groups whose proper closed subgroups are all discrete sorry line 59 source
Source documentation
Describe all minimal topological groups, that is, all non-discrete Hausdorff topological groups whose proper closed subgroups are all discrete.

depends on sorry what this means
Statement uses: Group, TopologicalSpace, Iff, Kourovka.«1.74».IsMinimalTopologicalGroup, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem kourovka_1_74 :
    ∀ (G : Type) [Group G] [TopologicalSpace G],
      IsMinimalTopologicalGroup G ↔
        (answer(sorry) : ∀ (G : Type) [Group G] [TopologicalSpace G], Prop) G := by
…
```
Mentions: Kourovka.«1.74».IsMinimalTopologicalGroup
def Kourovka.«1.74».IsTarskiMonster A Tarski monster group: an infinite group in which every non-trivial proper subgroup has order a fixed prime $p$ line 69 source
Source documentation
A Tarski monster group: an infinite group in which every non-trivial proper subgroup has order a fixed prime $p$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group
Proof uses: And, Infinite, Exists, Nat, Nat.Prime, Subgroup, Ne, Bot.bot, Top.top, Eq, Nat.card, Subtype, Membership.mem
4 instances, coercions and auxiliary constants left out.
```
def IsTarskiMonster (G : Type*) [Group G] : Prop :=
…
```
theorem Kourovka.«1.74».kourovka_1_74.variants.tarski_monster There exists a Tarski monster group that admits a non-discrete Hausdorff group topology sorry line 81 source
Source documentation
There exists a Tarski monster group that admits a non-discrete Hausdorff group topology. This follows from Theorem 1.4 of Klyachko, Olshanskii and Osin, which gives a topologizable Tarski monster of every sufficiently large odd exponent $n$; taking $n$ to be a prime $p$ makes every non-trivial proper subgroup cyclic of order $p$. Any such group is a minimal topological group.

depends on sorry what this means
Statement uses: Iff, True, Exists, Group, TopologicalSpace, And, Kourovka.«1.74».IsTarskiMonster, IsTopologicalGroup, T2Space, Not, DiscreteTopology
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem kourovka_1_74.variants.tarski_monster : answer(True) ↔
    ∃ (G : Type) (_ : Group G) (_ : TopologicalSpace G),
      IsTarskiMonster G ∧ IsTopologicalGroup G ∧ T2Space G ∧
      ¬ DiscreteTopology G := by
…
```
Mentions: Kourovka.«1.74».IsTarskiMonster