proposition: Lemma char_abelianP : reflect (forall i : Iirr G, 'chi_i \is a linear_char) (abelian G).
proof: Proof. apply: (iffP idP) => [cGG i | CF_G]. rewrite qualifE/= irr_char /= irr1_degree. by rewrite irr_degree_abelian //; last apply: groupC. rewrite card_classes_abelian -NirrE -eqC_nat -irr_sum_square //. rewrite -{1}[Nirr G]card_ord -sumr_const; apply/eqP/eq_bigr=> i _. by rewrite lin_char1 ?expr1n ?CF_G. 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)