ConjugacyClassSizes.lean`

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

8 declarations, 2 contain sorry, 0 declared as axioms

The $S_3$-conjecture (conjugacy classes of distinct sizes)

References:

W. Zhou, I. Gorshkov, On $\{2,3,5\}$-groups with conjugacy classes of distinct sizes, arXiv:2606.22244 (2026).
F. M. Markel, Groups with many conjugate elements, J. Algebra 26 (1973), 69–74. (Origin of the $S_3$-conjecture.)
R. Knörr, W. Lempken, B. Thielcke, The $S_3$-conjecture for solvable groups, Israel J. Math. 91 (1995), 61–76.
J. Zhang, Finite groups with many conjugate elements, J. Algebra 170 (1994), 608–624.
Z. Arad, M. Muzychuk, A. Oliver, On groups with conjugacy classes of distinct sizes, J. Algebra 280 (2004), 537–576.
Conjugacy class

A finite group in which distinct conjugacy classes have distinct cardinalities is called an anti-homogeneous group (or ah-group). The symmetric group $S_3$ is an ah-group: its three conjugacy classes have sizes $1$, $2$, and $3$. Markel's $S_3$-conjecture (1973) asserts that, up to isomorphism, $S_3$ is the only nontrivial finite ah-group. The conjecture has been proved for all solvable groups (independently by Zhang and by Knörr–Lempken–Thielcke), but the general non-solvable case remains open.

Imports (1)

FormalConjecturesUtil

def ConjugacyClassSizes.conjClassCard The cardinality of the conjugacy class c line 52 source
Source documentation
The cardinality of the conjugacy class c.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Monoid, ConjClasses, Nat
Proof uses: Nat.card, ConjClasses.carrier
1 instances, coercions and auxiliary constants left out.
```
noncomputable def conjClassCard {G : Type*} [Monoid G] (c : ConjClasses G) : ℕ :=
…
```
Mentions: c
def ConjugacyClassSizes.HasDistinctConjClassSizes Documented declaration: HasDistinctConjClassSizes line 61 source
Source documentation
A finite group G satisfies HasDistinctConjClassSizes if the map assigning to each conjugacy class the cardinality of its carrier is injective, i.e. distinct conjugacy classes have distinct sizes.
Such a group is called an anti-homogeneous group (or ah-group).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype
Proof uses: Function.Injective, ConjClasses, Nat, ConjugacyClassSizes.conjClassCard
2 instances, coercions and auxiliary constants left out.
```
def HasDistinctConjClassSizes (G : Type*) [Group G] [Fintype G] : Prop :=
…
```
theorem ConjugacyClassSizes.hasDistinctConjClassSizes_iff Equivalently, two conjugacy classes with the same cardinality must coincide line 68 source
Source documentation
Equivalently, two conjugacy classes with the same cardinality must coincide.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Iff, ConjugacyClassSizes.HasDistinctConjClassSizes, ConjClasses, Eq, Nat, ConjugacyClassSizes.conjClassCard
Proof uses: of_eq_true, iff_self
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hasDistinctConjClassSizes_iff {G : Type*} [Group G] [Fintype G] :
    HasDistinctConjClassSizes G ↔
      ∀ (a b : ConjClasses G), conjClassCard a = conjClassCard b → a = b := by
…
```
Mentions: ConjugacyClassSizes.HasDistinctConjClassSizes, ConjugacyClassSizes.conjClassCard
theorem ConjugacyClassSizes.hasDistinctConjClassSizes_of_subsingleton The trivial group is anti-homogeneous, since it has a single conjugacy class line 77 source
Source documentation
The trivial group is anti-homogeneous, since it has a single conjugacy class.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Subsingleton, ConjugacyClassSizes.HasDistinctConjClassSizes
Proof uses: ConjClasses, Eq.mpr, Eq, Nat, ConjugacyClassSizes.conjClassCard, id, congrArg, propext, ConjugacyClassSizes.hasDistinctConjClassSizes_iff, Subsingleton.elim
3 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem hasDistinctConjClassSizes_of_subsingleton {G : Type*} [Group G] [Fintype G]
    [Subsingleton G] : HasDistinctConjClassSizes (G := G) := by
…
```
Mentions: ConjugacyClassSizes.HasDistinctConjClassSizes
theorem ConjugacyClassSizes.trivial_center_of_hasDistinctConjClassSizes An anti-homogeneous group has trivial center, since each central element is in its own conjugacy class line 89 source
Source documentation
An anti-homogeneous group has trivial center, since each central element is in its own conjugacy class.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, ConjugacyClassSizes.HasDistinctConjClassSizes, Eq, Subgroup, Subgroup.center, Bot.bot
Proof uses: Membership.mem, Nat, ConjugacyClassSizes.conjClassCard, ConjClasses.mk, OfNat.ofNat, Set, ConjClasses.carrier, Singleton.singleton, Set.ext, Eq.mpr, Iff, ConjClasses, id, congrArg, propext, ConjClasses.mem_carrier_iff_mk_eq, IsConj, ConjClasses.mk_eq_mk_iff_isConj, Set.mem_singleton_iff, Iff.intro and 24 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
```
theorem trivial_center_of_hasDistinctConjClassSizes {G : Type*} [Group G] [Fintype G]
    (h : HasDistinctConjClassSizes G) : Subgroup.center G = ⊥ := by
…
```
Mentions: ConjugacyClassSizes.HasDistinctConjClassSizes
theorem ConjugacyClassSizes.hasDistinctConjClassSizes_perm_fin_three The symmetric group $S_3$ is anti-homogeneous, since its three conjugacy classes have sizes $1$, $2$ and $3$ line 119 source
Source documentation
The symmetric group $S_3$ is anti-homogeneous, since its three conjugacy classes have sizes $1$, $2$ and $3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ConjugacyClassSizes.HasDistinctConjClassSizes, Equiv.Perm, Fin, OfNat.ofNat, Nat
Proof uses: Eq, ConjClasses, ConjugacyClassSizes.conjClassCard, Finset.card, Set.toFinset, ConjClasses.carrier, funext, Eq.mpr, Nat.card, id, congrArg, ConjugacyClassSizes.conjClassCard.eq_1, Fintype.card, Nat.card_eq_fintype_card, Eq.symm, Set.toFinset_card, Eq.refl, Function.Injective, of_decide_eq_true, Bool and 2 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem hasDistinctConjClassSizes_perm_fin_three :
    HasDistinctConjClassSizes (Equiv.Perm (Fin 3)) := by
…
```
Mentions: ConjugacyClassSizes.HasDistinctConjClassSizes
theorem ConjugacyClassSizes.conjClassSizes_iff_sym_three Markel's $S_3$-conjecture (1973): any nontrivial finite ah-group is isomorphic to $S_3$ sorry line 135 source
Source documentation
Markel's $S_3$-conjecture (1973): any nontrivial finite ah-group is isomorphic to $S_3$.
The conjecture is open in general; it is known to be true for solvable groups.

depends on sorry what this means
Statement uses: Group, Fintype, Nontrivial, ConjugacyClassSizes.HasDistinctConjClassSizes, Nonempty, MulEquiv, Equiv.Perm, Fin, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjClassSizes_iff_sym_three
    (G : Type) [Group G] [Fintype G] [Nontrivial G]
    (h : HasDistinctConjClassSizes (G := G)) :
…
```
Mentions: ConjugacyClassSizes.HasDistinctConjClassSizes
theorem ConjugacyClassSizes.conjClassSizes_iff_sym_three_solvable The $S_3$-conjecture holds for solvable groups: every nontrivial solvable finite ah-group is isomorphic to $S_3$ sorry line 147 source
Source documentation
The $S_3$-conjecture holds for solvable groups: every nontrivial solvable finite ah-group is isomorphic to $S_3$. This was proved independently by Zhang (1994) and by Knörr–Lempken–Thielcke (1995).

depends on sorry what this means
Statement uses: Group, Fintype, Group.IsSolvable, Nontrivial, ConjugacyClassSizes.HasDistinctConjClassSizes, Nonempty, MulEquiv, Equiv.Perm, Fin, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjClassSizes_iff_sym_three_solvable
    (G : Type) [Group G] [Fintype G] [Group.IsSolvable G] [Nontrivial G]
    (h : HasDistinctConjClassSizes (G := G)) :
…
```
Mentions: ConjugacyClassSizes.HasDistinctConjClassSizes