proposition: Proposition stable_factor_cent gT (A G H : {group gT}) : A \subset 'C(H) -> stable_factor A H G -> coprime #|G| #|A| -> solvable G -> A \subset 'C(G).
proof: Proof. move=> cHA /and3P[sRH sHG nHG] coGA solG. suffices: G \subset 'C_G(A) by rewrite subsetI subxx centsC. rewrite -(quotientSGK nHG) ?subsetI ?sHG 1?centsC //. by rewrite coprime_quotient_cent ?cents_norm ?subsetI ?subxx ?quotient_cents2r. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun eqtype ssrnat seq path div fintype) mathcomp(bigop prime binomial finset fingroup morphism perm automorphism) mathcomp(quotient action gproduct gfunctor commutator) mathcomp(ssralg finalg zmodp cyclic center pgroup finmodule gseries) mathcomp(nilpotent sylow abelian maximal hall extremal) mathcomp(matrix mxalgebra mxrepresentation mxabelem)