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.

noncomputable def ConjugacyClassSizes.conjClassCard {G : Type u_1} [Monoid G] (c : ConjClasses G) : ℕ

The cardinality of the conjugacy class c.

Equations

• ConjugacyClassSizes.conjClassCard c = Nat.card ↑c.carrier

def ConjugacyClassSizes.HasDistinctConjClassSizes (G : Type u_1) [Group G] [Fintype G] : Prop

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).

Equations

• ConjugacyClassSizes.HasDistinctConjClassSizes G = Function.Injective ConjugacyClassSizes.conjClassCard

theorem ConjugacyClassSizes.hasDistinctConjClassSizes_iff {G : Type u_1} [Group G] [Fintype G] : HasDistinctConjClassSizes G ↔ ∀ (a b : ConjClasses G), conjClassCard a = conjClassCard b → a = b

Equivalently, two conjugacy classes with the same cardinality must coincide.

theorem ConjugacyClassSizes.hasDistinctConjClassSizes_of_subsingleton {G : Type u_1} [Group G] [Fintype G] [Subsingleton G] : HasDistinctConjClassSizes G

The trivial group is anti-homogeneous, since it has a single conjugacy class.

theorem ConjugacyClassSizes.trivial_center_of_hasDistinctConjClassSizes {G : Type u_1} [Group G] [Fintype G] (h : HasDistinctConjClassSizes G) : Subgroup.center G = ⊥

An anti-homogeneous group has trivial center, since each central element is in its own conjugacy class.

theorem ConjugacyClassSizes.hasDistinctConjClassSizes_perm_fin_three : HasDistinctConjClassSizes (Equiv.Perm (Fin 3))

The symmetric group $S_3$ is anti-homogeneous, since its three conjugacy classes have sizes $1$, $2$ and $3$.

theorem ConjugacyClassSizes.conjClassSizes_iff_sym_three (G : Type) [Group G] [Fintype G] [Nontrivial G] (h : HasDistinctConjClassSizes G) : Nonempty (G ≃* Equiv.Perm (Fin 3))

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.

theorem ConjugacyClassSizes.conjClassSizes_iff_sym_three_solvable (G : Type) [Group G] [Fintype G] [IsSolvable G] [Nontrivial G] (h : HasDistinctConjClassSizes G) : Nonempty (G ≃* Equiv.Perm (Fin 3))

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).