proposition: Lemma cfcenter_subset_center chi : ('Z(chi)%CF / cfker chi)%g \subset 'Z(G / cfker chi)%g.
proof: Proof. case Nchi: (chi \is a character); last first. by rewrite /cfcenter Nchi trivg_quotient sub1G. rewrite subsetI quotientS ?cfcenter_sub // quotient_cents2r //=. case/char_reprP: Nchi => rG ->{chi}; rewrite cfker_repr cfcenter_repr gen_subG. apply/subsetP=> _ /imset2P[x y /setIdP[Gx /is_scalar_mxP[c rGx]] Gy ->]. rewrite inE groupR //= !repr_mxM ?groupM ?groupV // rGx -(scalar_mxC c) -rGx. by rewrite !mulmxA !repr_mxKV. 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)