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Prescribed Product Composition

Abstract

In an arbitrary group, ordered corrected automorphism value sets concatenate through Fin-indexed tuples. The proof transports corrections through genuine inner automorphisms and retains target-independent correction choices. These helpers are consumed by the five-block intrinsic SU construction.

Definition 1.1 (Covers).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PrescribedProductComposition.Covers

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PrescribedProductComposition.Covers (✓ std3).

Source. Repository-derived.

Commentary.

One genuine right-inner correction is selected before all targets in T. The full prescribed tuple and original divisor powers are retained.

Theorem 1.2 (covers append).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PrescribedProductComposition.covers_append

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

Source. Repository-derived.

Commentary.

Consume two consecutive coverage blocks without reordering their values. The resulting correction still precedes every target of the set product.

Theorem 1.3 (covers of left inner).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PrescribedProductComposition.covers_of_left_inner

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

Source. Repository-derived.

Commentary.

A genuine left inner change of the supplied automorphisms can be absorbed into right-inner corrections, before any target is chosen.

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