proposition: Lemma rker_linear : n = 1 -> G^`(1)%g \subset rker.
proof: Proof. move=> n1; rewrite gen_subG; apply/subsetP=> xy; case/imset2P=> x y Gx Gy ->. rewrite !inE groupR //= /commg mulgA -invMg repr_mxM ?groupV ?groupM //. rewrite mulmxA (can2_eq (repr_mxK _) (repr_mxKV _)) ?groupM //. rewrite !repr_mxV ?repr_mxM ?groupM //; move: (rG x) (rG y). by rewrite n1 => rx ry; rewrite (mx11_scalar rx) scalar_mxC. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime) mathcomp(ssralg poly polydiv finset fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup matrix mxalgebra) mathcomp(mxpoly)