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Unitary Sylow Roots

Abstract

Actual unitary sylow roots supplies the parameterized matrix, finite-field or ordered-product identities consumed by the whole-SU K5 width and intrinsic prescribed-product proof paths. All original hypotheses and actual carriers are retained; this source provides no oracle for bare automorphism recognition.

Theorem 1.1 (steinberg transvection).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.steinberg_transvection

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.steinberg_transvection (✓ std3). ∎

Source. Repository-derived.

Commentary.

The genuine field/graph action on actual SL transvections.

Theorem 1.2 (transvections commute).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.transvections_commute

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.transvections_commute (✓ std3). ∎

Source. Repository-derived.

Commentary.

Two actual transvections with disjoint multiplication positions commute.

Theorem 1.3 (paired root fixed).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.paired_root_fixed

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.paired_root_fixed (✓ std3). ∎

Source. Repository-derived.

Commentary.

A paired upper root away from the middle of the anti-diagonal.

Theorem 1.4 (paired root matrix).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.paired_root_matrix

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.paired_root_matrix (✓ std3). ∎

Source. Repository-derived.

Commentary.

Literal matrix of the paired root, with no hidden root-group assumptions.

Theorem 1.5 (central root fixed).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.central_root_fixed

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.central_root_fixed (✓ std3). ∎

Source. Repository-derived.

Commentary.

Central long root, including characteristic two.

Definition 1.6 (shortRoot).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot (✓ std3).

Source. Repository-derived.

Commentary.

The actual three-entry short root is a product of determinant-one transvections.

Theorem 1.7 (shortRoot matrix).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot_matrix

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot_matrix (✓ std3). ∎

Source. Repository-derived.

Commentary.

The parameterized shortRoot matrix statement describes the actual unitary geometry or prescribed ordered-product construction. Its Lean telescope retains every field, rank, involution, input and equality hypothesis shown.

Theorem 1.8 (shortRoot inverse matrix).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot_inverse_matrix

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot_inverse_matrix (✓ std3). ∎

Source. Repository-derived.

Commentary.

Explicit inverse matrix of the actual short-root product.

Theorem 1.9 (shortRoot fixed).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot_fixed

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowRoots.shortRoot_fixed (✓ std3). ∎

Source. Repository-derived.

Commentary.

Actual short-root fixedness follows from the trace equation.

The results concern special-unitary matrix groups over finite fields. Their geometric setting is Nikolov and Segal, On finitely generated profinite groups, II: Products in quasisimple groups, Annals of Mathematics 165 (2007), 239-273, DOI 10.4007/annals.2007.165.239. Prescribed-product statements retain their field-size, rank and automorphism hypotheses.

References