DiameterSimpleFiniteGroups.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

7 declarations, 2 contain sorry, 0 declared as axioms

Babai–Seress Conjectures on the Diameter of Finite Groups

References:

This file contains two conjectures from the Babai–Seress paper:

Conjecture 1.5: $\operatorname{diam}(A_n) < n^C$ for some absolute constant $C$, where $A_n$ is the alternating group on $n$ elements.
Conjecture 1.7: $\operatorname{diam}(G) < (\log |G|)^C$ for some absolute constant $C$, where $G$ ranges over all non-abelian finite simple groups.

Conjecture 1.7 generalises Conjecture 1.5, since for $G = A_n$ we have $\log |A_n| \approx n \log n$, so a polylogarithmic bound in $|G|$ implies a polynomial bound in $n$.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def BabaiSeressConjectures.cayleyGraph The (undirected) Cayley graph of a group $G$ with respect to a generating set $S$ line 55 source
Source documentation
The (undirected) Cayley graph of a group $G$ with respect to a generating set $S$. Two elements $g, h \in G$ are adjacent iff $g \neq h$ and $g^{-1} h \in S$ or $h^{-1} g \in S$.
This is constructed using SimpleGraph.fromRel, which takes the relation $g \sim h \iff g^{-1} h \in S$ and automatically symmetrizes it (via disjunction with the reverse relation) and enforces irreflexivity (via $g \neq h$). In particular, this definition effectively uses the symmetrization $S \cup S^{-1}$, so it produces a standard undirected Cayley graph even when $S$ is not itself symmetric.

standard axioms: Quot.sound, propext what this means
Statement uses: Group, Set, SimpleGraph
Proof uses: SimpleGraph.fromRel, Membership.mem, HMul.hMul, Inv.inv
11 instances, coercions and auxiliary constants left out.
```
def cayleyGraph {G : Type*} [Group G] (S : Set G) : SimpleGraph G :=
…
```
def BabaiSeressConjectures.groupDiam The diameter of a finite group $G$, defined as the maximum diameter of the Cayley graphs $\Gamma(G, A)$ over all generating sets $A$ of $G$ line 61 source
Source documentation
The diameter of a finite group $G$, defined as the maximum diameter of the Cayley graphs $\Gamma(G, A)$ over all generating sets $A$ of $G$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Set, And, Eq, Subgroup, Subgroup.closure, Top.top, SimpleGraph.diam, BabaiSeressConjectures.cayleyGraph
2 instances, coercions and auxiliary constants left out.
```
noncomputable def groupDiam (G : Type*) [Group G] [Fintype G] : ℕ :=
…
```
theorem BabaiSeressConjectures.groupDiam_fin_one For the trivial group (with one element), the group diameter is zero, since every Cayley graph has only one vertex and hence diameter zero line 67 source
Source documentation
For the trivial group (with one element), the group diameter is zero, since every Cayley graph has only one vertex and hence diameter zero.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, BabaiSeressConjectures.groupDiam, Subtype, Equiv.Perm, Fin, OfNat.ofNat, Membership.mem, Subgroup, alternatingGroup
Proof uses: id, Iff.mp, LE.le, SupSet.sSup, Set.ofPred, Exists, Set, And, Subgroup.closure, Top.top, SimpleGraph.diam, BabaiSeressConjectures.cayleyGraph, Nat.le_zero, csSup_le, Exists.intro, Set.univ, And.intro, Subgroup.closure_univ, Iff.mpr, Or and 7 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem groupDiam_fin_one : groupDiam (alternatingGroup (Fin 0)) = 0 := by
…
```
Mentions: BabaiSeressConjectures.groupDiam
theorem BabaiSeressConjectures.groupDiam_alternating_three The alternating group line 80 source
Source documentation
The alternating group $A_3 \cong \mathbb{Z}/3\mathbb{Z}$ has group diameter $1$: every non-trivial generating set produces a complete Cayley graph $K_3$, since any single non-identity element and its inverse already reach the entire group.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, BabaiSeressConjectures.groupDiam, Subtype, Equiv.Perm, Fin, OfNat.ofNat, Membership.mem, Subgroup, alternatingGroup
Proof uses: Nontrivial, Iff.mp, LT.lt, Fintype.card, Fintype.one_lt_card_iff_nontrivial, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Set, Subgroup.closure, Top.top, SimpleGraph, BabaiSeressConjectures.cayleyGraph, Eq.mpr, Ne, SimpleGraph.Adj, congrArg and 58 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 36 steps
Open to load the checked before-and-after goal states.
```
theorem groupDiam_alternating_three : groupDiam (alternatingGroup (Fin 3)) = 1 := by
…
```
Mentions: BabaiSeressConjectures.groupDiam
theorem BabaiSeressConjectures.groupDiam_perm_two The symmetric group line 119 source
Source documentation
The symmetric group $S_2 \cong \mathbb{Z}/2\mathbb{Z}$ has group diameter $1$: the unique generating set $\{(01)\}$ produces the complete graph $K_2$, since the single non-identity element and its inverse (which are equal) cover the only other vertex.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, BabaiSeressConjectures.groupDiam, Equiv.Perm, Fin, OfNat.ofNat
Proof uses: Nontrivial, Iff.mp, LT.lt, Fintype.card, Fintype.one_lt_card_iff_nontrivial, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Set, Subgroup, Subgroup.closure, Top.top, SimpleGraph, BabaiSeressConjectures.cayleyGraph, Eq.mpr, Ne, SimpleGraph.Adj and 60 more
94 instances, coercions and auxiliary constants left out.

Tactic trace: 36 steps
Open to load the checked before-and-after goal states.
```
theorem groupDiam_perm_two : groupDiam (Equiv.Perm (Fin 2)) = 1 := by
…
```
Mentions: BabaiSeressConjectures.groupDiam
theorem BabaiSeressConjectures.babai_seress_conjecture_alternating Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant sorry line 156 source
Source documentation
Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant $C$ such that the diameter of the alternating group $A_n$ satisfies $$\operatorname{diam}(A_n) \leq n^C.$$
Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.5

L4DEL explanation
Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant $C$ such that the diameter of the alternating group $A_n$ satisfies $$\operatorname{diam}(A_n) \leq n^C.$$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.580029-0)
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Exists, Nat, LE.le, Real, BabaiSeressConjectures.groupDiam, Subtype, Equiv.Perm, Fin, Membership.mem, Subgroup, alternatingGroup, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem babai_seress_conjecture_alternating :
    ∃ C : ℕ, ∀ n : ℕ,
    (groupDiam (alternatingGroup (Fin n)) : ℝ) ≤ (n : ℝ) ^ C := by
…
```
Mentions: BabaiSeressConjectures.groupDiam
theorem BabaiSeressConjectures.babai_seress_conjecture Babai–Seress Conjecture (Conjecture 1.7): There exists an absolute constant sorry line 168 source
Source documentation
Babai–Seress Conjecture (Conjecture 1.7): There exists an absolute constant $C$ such that every finite simple non-abelian group $G$ satisfies $$\operatorname{diam}(G) \leq (\log |G|)^C.$$
Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.7

L4DEL explanation
Babai–Seress Conjecture (Conjecture 1.7): There exists an absolute constant $C$ such that every finite simple non-abelian group $G$ satisfies $$\operatorname{diam}(G) \leq (\log |G|)^C.$$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.780029-0)
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Exists, Nat, Group, Fintype, IsSimpleGroup, Ne, HMul.hMul, LE.le, Real, BabaiSeressConjectures.groupDiam, HPow.hPow, Real.log, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem babai_seress_conjecture :
    ∃ C : ℕ,
    ∀ (G : Type) [Group G] [Fintype G] [IsSimpleGroup G],
    (∃ a b : G, a * b ≠ b * a) →
    (groupDiam G : ℝ) ≤ (Real.log (Fintype.card G : ℝ)) ^ C := by
…
```
Mentions: BabaiSeressConjectures.groupDiam