Part II SL2GeneralLinear
Abstract
Part II SL2GeneralLinear.
Theorem 1.1 (actual GL2 semilinear scalar product).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/PartIISL2GeneralLinear.actual_GL2_semilinear_scalar_product
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/PartIISL2GeneralLinear.actual_GL2_semilinear_scalar_product (✓ std3). ∎
Source. Repository-derived.
Commentary.
Conjugation by any invertible two-by-two matrix, followed by an arbitrary field automorphism, satisfies the actual SL2 scalar PRODUCT interface under the same bounds. Factoring the matrix into a determinant-one factor and a diagonal factor constructs the genuine inner and diagonal actions.
Nikolov and Segal, On finitely generated profinite groups, I: Strong completeness and uniform bounds, Annals of Mathematics 165 (2007), 171-238, DOI 10.4007/annals.2007.165.171, Section 10, Proposition 10.2 and equations (45)-(50) (pages 228-232). Part II: Products in quasisimple groups, Annals of Mathematics 165 (2007), 239-273, DOI 10.4007/annals.2007.165.239, Theorem 1.2, Lemma 4.1 (pages 247-248), and Lemma 7.1(a) with its A1 application (pages 257-261). These are formal adaptations and consequences of published mathematics, with explicit matrix, fixed-field and Sylow arguments. No originality claim or redistribution of the papers is made. Arbitrary automorphisms are handled for the A1 family; other finite-simple families, the full uniform scalar supplier, quasisimple central covers, all-length Proposition 10.2, uniform width and restricted Burnside bounds, and full strong completeness remain open.