proposition: Lemma cap_cfcenter_irr : \bigcap_i 'Z('chi[G]_i)%CF = 'Z(G).
proof: Proof. apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) /= => [|i _]; last first. rewrite -(quotientSGK _ (normal_sub (cfker_center_normal _))). by rewrite cfcenter_eq_center morphim_center. by rewrite subIset // normal_norm // cfker_normal. set Z := \bigcap_i _. have sZG: Z \subset G by rewrite (bigcap_min 0) ?cfcenter_sub. rewrite subsetI sZG (sameP commG1P trivgP) -(TI_cfker_irr G). apply/bigcapsP=> i _; have nKiG := normal_norm (cfker_normal 'chi_i). rewrite -quotient_cents2 ?(subset_trans sZG) //. rewrite (subset_trans (quotientS _ (bigcap_inf i _))) //. by rewrite cfcenter_eq_center subsetIr. 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)