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Unitary Positive Sylow

Abstract

Actual unitary positive sylow 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 (upper pow).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryPositiveSylow.upper_pow

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

Source. Repository-derived.

Commentary.

The parameterized upper pow 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 1.2 (upper pow diag).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryPositiveSylow.upper_pow_diag

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

Source. Repository-derived.

Commentary.

The parameterized upper pow diag 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 1.3 (unitriangular of p power).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryPositiveSylow.unitriangular_of_p_power

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

Source. Repository-derived.

Commentary.

Any actual upper triangular p-power-order SL matrix in defining characteristic has diagonal one, by the Frobenius power identity.

Theorem 1.4 (positiveU maximal).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryPositiveSylow.positiveU_maximal

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

Source. Repository-derived.

Commentary.

Actual positive unitary U is maximal among p-subgroups. The proof uses its proved intrinsic normalizer and the native normalizer condition for finite p-groups; it assumes neither the unitary group order nor an index.

Definition 1.5 (positiveUSylow).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryPositiveSylow.positiveUSylow

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryPositiveSylow.positiveUSylow (✓ std3).

Source. Repository-derived.

Commentary.

Genuine native Sylow with exactly the actual positive-unitriangular SU carrier.

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