proposition: Lemma cfker_reg_quo G H : H <| G -> cfker (cfReg (G / H)%g %% H) = H.
proof: Proof. move=> nsHG; have [sHG nHG] := andP nsHG. apply/setP=> x; rewrite cfkerEchar ?cfMod_char ?cfReg_char //. rewrite -[in RHS in _ = RHS](setIidPr sHG) !inE; apply: andb_id2l => Gx. rewrite !cfModE // !cfRegE // morph1 eqxx. rewrite (sameP eqP (kerP _ (subsetP nHG x Gx))) ker_coset. by rewrite -!mulrnA eqr_nat eqn_pmul2l ?cardG_gt0 // (can_eq oddb) eqb_id. 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)