feat: add six finite-group-theory benchmark problems#64
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This PR adds six benchmark statements at the boundary of finite group theory and representation theory: Brauer's theorem on cyclotomic character values, an existential M_23 tensor-square problem, and four CFSG-adjacent results (Frobenius's kernel-is-normal theorem, Glauberman's Z*, Schreier's conjecture, and the order-restriction part of the classification of 5-transitive permutation groups). Each statement was checked adversarially before commit: the M_23 problem pins the diagonal action via an explicit equation rather than a sorry'd helper instance, the 5-transitive disjunction has 5 ≤ |X| to defeat vacuity and 7 ≤ n on the n!/2 branch, the Schreier file proves normality of Inn(S) ⊴ Aut(S) so the quotient typechecks, and the Brauer docstring is honest that this is the character-value statement (strictly weaker than the splitting-field statement). 🤖 Prepared with Claude Code Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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This PR adds six benchmark statements at the boundary of finite group theory and representation theory: Brauer's theorem on cyclotomic character values, an existential M₂₃ tensor-square problem in the same style as the existing exceptional-Lie tensor-square problems, and four CFSG-adjacent results — Frobenius's kernel-is-normal theorem, Glauberman's Z*, Schreier's conjecture, and the order-restriction part of the classification of 5-transitive permutation groups.
Each statement was checked adversarially before commit. Notable choices:
brauer_character_in_cyclotomic): asserts∃ φ : ℚ(ζ_exp G) →+* ℂ, ∀ g, tr(ρ g) ∈ φ.range. The docstring is explicit that this is the character-value form, which is strictly weaker than the full splitting-field statement (every irrep has a ℚ(ζₙ)-form). The latter would need scalar-extension scaffolding mathlib doesn't expose cleanly.m23_irrep_tensor_square_decomp): existential for a finite group of order 10 200 960 with a 22-dim irrep whose tensor square has 4 isotypic components. BothMonoidAlgebra ℂ Gactions (on V and on V ⊗ V) are explicit binders withIsScalarTower, and the V ⊗ V action is pinned by the diagonal-action equationg • (v ⊗ w) = (g • v) ⊗ (g • w)forg : G. The statement does not pin M₂₃ uniquely, but constructing any witness is essentially equivalent to constructing M₂₃.frobenius_kernel_isNormal): explicit2 ≤ |X|to make the Frobenius-action package fully nondegenerate.glauberman_zStar): uses the global isolated-involution form (no distinct conjugate oftcommutes witht); the conclusion form∀ g, g·t·g⁻¹·t⁻¹ ∈ Nis exactly "t central in G/N".schreier_conjecture): a localinnNormalinstance provesInn(S) ⊴ Aut(S)(one-linesimpafterext) so the quotientMulAut S ⧸ MulAut.conj.rangetypechecks.five_transitive_card_classification):5 ≤ |X|defeats vacuity (otherwiseC₃onFin 3qualifies); then!/2branch is gated by7 ≤ nsince Aₙ is only 5-transitive forn ≥ 7. Title is "Possible orders of …" rather than "Classification of …" since only orders are formalized.🤖 Prepared with Claude Code