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Unitary Sylow Carrier

Abstract

Actual unitary sylow carrier 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.

Definition 1.1 (specialUnitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowCarrier.specialUnitary

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowCarrier.specialUnitary (✓ std3).

Source. Repository-derived.

Commentary.

Literal determinant-one anti-diagonal unitary group: fixed points of the accepted field/inverse-transpose Steinberg automorphism on actual SL.

Definition 1.2 (positiveU).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowCarrier.positiveU

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowCarrier.positiveU (✓ std3).

Source. Repository-derived.

Commentary.

Exactly the literal positive unitriangular matrices in actual SU.

Theorem 1.3 (unitriangular iff depth).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowCarrier.unitriangular_iff_depth

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

Source. Repository-derived.

Commentary.

The parameterized unitriangular iff depth 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 1.4 (mem positiveU iff).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowCarrier.mem_positiveU_iff

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

Source. Repository-derived.

Commentary.

The parameterized mem positiveU iff 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 1.5 (positiveU isPGroup).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowCarrier.positiveU_isPGroup

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

Source. Repository-derived.

Commentary.

P-group recognition reuses the accepted actual SLn theorem by injection.

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