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Twisted values generate a noncommutative simple group

Abstract

Twisted values generate a noncommutative simple group.

Theorem 1.1 (ordinary from twisted).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/SmallTwistedGeneration.ordinary_from_twisted

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/SmallTwistedGeneration.ordinary_from_twisted (✓ std3). ∎

Source. Repository-derived.

Commentary.

Three actual twisted values express an ordinary commutator, with arbitrary automorphisms a and b and without a coverage oracle.

Theorem 1.2 (noncommutative simple commutator top).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/SmallTwistedGeneration.noncommutative_simple_commutator_top

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/SmallTwistedGeneration.noncommutative_simple_commutator_top (✓ std3). ∎

Source. Repository-derived.

Commentary.

The real generated commutator subgroup of a noncommutative simple group is the whole group.

Theorem 1.3 (twisted closure top).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/SmallTwistedGeneration.twisted_closure_top

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/SmallTwistedGeneration.twisted_closure_top (✓ std3). ∎

Source. Repository-derived.

Commentary.

Every actual twisted value set generates the whole noncommutative simple group.

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.

References