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Unitary Sylow Normalizer

Abstract

Actual unitary sylow normalizer 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 (normalizer preserves prefix).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowNormalizer.normalizer_preserves_prefix

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

Source. Repository-derived.

Commentary.

Every prefix is preserved by a normalizer element, from the intrinsic unitary-root characterization rather than an assumed fixed flag.

Theorem 1.2 (upper of normalizer).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowNormalizer.upper_of_normalizer

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

Source. Repository-derived.

Commentary.

The genuine SU normalizer has literal zero entries below the diagonal.

Theorem 1.3 (mem normalizer iff upper).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowNormalizer.mem_normalizer_iff_upper

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

Source. Repository-derived.

Commentary.

Exact normalizer recognition for actual positive unitary U.

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