proposition: Proposition coprime_cent_prod gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> solvable G -> [~: G, A] * 'C_G(A) = G.
proof: Proof. move=> nGA; have sRG: [~: G, A] \subset G by rewrite commg_subl. rewrite -(Lagrange sRG) coprimeMl => /andP[coRA _] /(solvableS sRG). exact: coprimeR_cent_prod. 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)