proposition: Lemma normal_sub_max_pgroup pi H M G : [max M | pi.-subgroup(G) M] -> pi.-group H -> H <| G -> H \subset M.
proof: Proof. move=> maxM piH /andP[sHG nHG]. apply: comm_sub_max_pgroup piH sHG _ => //; apply: commute_sym; apply: normC. by apply: subset_trans nHG; case/andP: (maxgroupp maxM). Qed.
imports: mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq div) mathcomp(fintype bigop finset prime fingroup morphism) mathcomp(gfunctor automorphism quotient action gproduct) mathcomp(cyclic)