Strict growth of finite product sets
Abstract
Strict growth of finite product sets.
Theorem 1.1 (finite right stable inv).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/FiniteGeneratingGrowth.finite_right_stable_inv
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/FiniteGeneratingGrowth.finite_right_stable_inv (✓ std3). ∎
Source. Repository-derived.
Commentary.
A finite set stable under right multiplication by g is also stable under g inverse, by actual finite injectivity and surjectivity.
Theorem 1.2 (card mul strict of generating).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/FiniteGeneratingGrowth.card_mul_strict_of_generating
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/FiniteGeneratingGrowth.card_mul_strict_of_generating (✓ std3). ∎
Source. Repository-derived.
Commentary.
For a nonempty proper finite set A and a finite generating set T containing one, the actual product set A*T has strictly larger cardinality.
Nikolov and Segal, On finitely generated profinite groups, I: Strong completeness and uniform bounds, Annals of Mathematics 165 (2007), 171-238, DOI 10.4007/annals.2007.165.171, Section 8, Lemma 8.3 and Proposition 8.4, Section 9, Proposition 9.1 (pages 223-226), and Section 10, Proposition 10.2 and equations (45)-(50) (pages 228-232). Part II: Products in quasisimple groups, Annals of Mathematics 165 (2007), 239-273, DOI 10.4007/annals.2007.165.239, Theorem 1.2 and Lemma 4.1 (pages 247-248). These are formal adaptations and conditional consequences of published mathematics, with elementary finite-set growth supporting bounded-small coverage; no originality claim or redistribution of the papers is made. Large-simple scalar PRODUCT existence, quasisimple central covers, all-length Proposition 10.2, uniform width and restricted Burnside bounds, and full strong completeness remain unproved.