Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Algebra P-paper-conjugacy-class-sizes

The S_3-conjecture (conjugacy classes of distinct sizes)

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.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Start working on it Submit a claim Follow
Cite 
@misc{cairn-paper-conjugacy-class-sizes,
  title        = {The S_3-conjecture (conjugacy classes of distinct sizes)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-conjugacy-class-sizes}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Markel's -conjecture (1973): any nontrivial finite ah-group is isomorphic to .

The conjecture is open in general; it is known to be true for solvable groups.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.ConjugacyClassSizes.

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

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • W. Zhou, I. Gorshkov, On -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 -conjecture.)
  • R. Knörr, W. Lempken, B. Thielcke, The -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 is an ah-group: its three conjugacy classes have sizes , , and . Markel's -conjecture (1973) asserts that, up to isomorphism, 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.

Source and licence

Imported from Formal Conjectures (research papers), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.