Lagrange theorem on S₄ subgroups
Theorem.Lagrange theorem on S₄ subgroups — H ≤ G ⇒ |H| divides |G|.
Proof.every subgroup order divides |G|: the subgroups of S₄ (order 24) have orders {1,2,3,4,6,8,12,24}, all dividing 24, enumerated by closure — the theorem underlying Cauchy, Sylow and cosets.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveThirtyTwo
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"Lagrange theorem on S₄ subgroups" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation discoveredTheoremsWaveThirtyTwo.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (discoveredTheoremsWaveThirtyTwo @ src/9/1) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes discoveredTheoremsWaveThirtyTwo (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.