BoundedBurnsideProblem.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

1 declarations, 1 contain sorry, 0 declared as axioms

Bounded Burnside problem

References:

Wikipedia
[NA68] P. S. Novikov, S. I. Adian. "Infinite periodic groups I-III." Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968).
[Ad79] S. I. Adian. The Burnside Problem and Identities in Groups. Springer, 1979.
[Iv94] S. V. Ivanov. "The free Burnside groups of sufficiently large exponents." Internat. J. Algebra Comput. 4 (1994), 1-308.
[Ly96] I. G. Lysënok. "Infinite Burnside groups of even exponent." Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), 3-224.

Imports (1)

FormalConjecturesUtil

theorem BoundedBurnsideProblem.bounded_burnside_problem Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$, $g^n = 1$ sorry line 54 source
Source documentation
Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$, $g^n = 1$. Must $G$ be finite?
The answer is negative. Novikov and Adian proved that for every odd $n > 4381$ there exist infinite, finitely generated groups of exponent $n$ [NA68]; Adian later reduced this to odd $n > 665$ [Ad79]. The even case is harder: Ivanov proved $B(m, n)$ infinite for $m > 1$ and even $n \ge 2^{48}$ divisible by $2^9$ [Iv94], and Lysënok improved this to $m > 1$ and $n \ge 8000$ [Ly96]. Any such group, for example the free Burnside group $B(2, 667)$, refutes the statement below.
Note this concerns only the universally quantified question stated here, which a single counterexample closes. The classification of which free Burnside groups $B(m, n)$ are finite remains open, with $B(2, 5)$ the best known open case.

depends on sorry what this means
Statement uses: Iff, False, Group, Group.FG, Nat, GT.gt, OfNat.ofNat, Eq, HPow.hPow, Finite
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem bounded_burnside_problem :
    answer(False) ↔ ∀ (G : Type) [Group G] (fin_gen : Group.FG G)
      (n : ℕ) (hn : n > 0) (bounded : ∀ g : G, g^n = 1), Finite G := by
…
```