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

formal-conjectures · paper · ams-20

**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.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/2606.22244
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Conjugacy_class
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Paper/ConjugacyClassSizes.lean
Local target: Corpus.PaperConjugacyClassSizes.conjClassSizes_iff_sym_three
Source SHA-256: 51e419bae232289dd7b715d4115aec773926c5913284c85b8feda2d2b3942e2a
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record