proposition: Lemma cfker_Ind chi : H \subset G -> chi \is a character -> chi != 0 -> cfker ('Ind[G, H] chi) = gcore (cfker chi) G.
proof: Proof. move=> sHG Nchi nzchi; rewrite !cfker_nzcharE ?cfInd_char ?cfInd_eq0 //. apply/setP=> x; rewrite inE cfIndE // (can2_eq (mulVKf _) (mulKf _)) ?neq0CG //. rewrite cfInd1 // mulrA -natrM Lagrange // mulr_natl -sumr_const. apply/eqP/bigcapP=> [/normC_sum_upper ker_chiG_x y Gy | ker_chiG_x]. by rewrite mem_conjg inE ker_chiG_x ?groupV // => z _; apply: char1_ge_norm. by apply: eq_bigr => y /groupVr/ker_chiG_x; rewrite mem_conjgV inE => /eqP. 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)