Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Combinatorics P-diameter-simple-finite-groups

Babai–Seress Conjectures on the Diameter of Finite Groups

Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant C such that the diameter of the alternating group A_n satisfies diam(A_n) ≤ n^C. Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.580029-0)

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.

Start working on it Submit a claim Follow
Cite 
@misc{cairn-diameter-simple-finite-groups,
  title        = {Babai–Seress Conjectures on the Diameter of Finite Groups},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/diameter-simple-finite-groups}},
  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

babai_seress_conjecture_alternating. Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant such that the diameter of the alternating group satisfies

Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.580029-0)

babai_seress_conjecture. Babai–Seress Conjecture (Conjecture 1.7): There exists an absolute constant such that every finite simple non-abelian group satisfies

Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.780029-0)

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.DiameterSimpleFiniteGroups (2 statements).

theorem babai_seress_conjecture_alternating :
    ∃ C : ℕ, ∀ n : ℕ,
    (groupDiam (alternatingGroup (Fin n)) : ℝ) ≤ (n : ℝ) ^ C
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

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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

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

  • Conjecture 1.5: for some absolute constant , where is the alternating group on elements.
  • Conjecture 1.7: for some absolute constant , where ranges over all non-abelian finite simple groups.

Conjecture 1.7 generalises Conjecture 1.5, since for we have , so a polylogarithmic bound in implies a polynomial bound in .

Source and licence

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