proposition: Lemma solvable_has_lin_char G : G :!=: 1%g -> solvable G -> exists2 i, 'chi[G]_i \is a linear_char & 'chi_i != 1.
proof: Proof. move=> ntG solG. suff /subsetPn[i]: ~~ ([pred i | 'chi[G]_i \is a linear_char] \subset pred1 0). by rewrite !inE -(inj_eq irr_inj) irr0; exists i. rewrite (contra (@subset_leq_card _ _ _)) // -ltnNge card1 card_lin_irr. by rewrite indexg_gt1 proper_subn // (sol_der1_proper solG). Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype choice ssrnat seq) mathcomp(path div fintype tuple finfun bigop prime order) mathcomp(ssralg poly finset gproduct fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup nilpotent sylow) mathcomp(abelian matrix mxalgebra mxpoly mxrepresentation) mathcomp(vector ssrnum algC classfun archimedean)