proposition: Remark sigma_mmax_exists p : p \in \pi(G) -> {M : {group gT} | M \in 'M & p \in \sigma(M)}.
proof: Proof. move=> piGp; have [P sylP] := Sylow_exists p [set: gT]. have ntP: P :!=: 1 by rewrite -rank_gt0 (rank_Sylow sylP) p_rank_gt0. have ltPG: P \proper G := mFT_pgroup_proper (pHall_pgroup sylP). have [M maxNM] := mmax_exists (mFT_norm_proper ntP ltPG). have{maxNM} [maxM sNM] := setIdP maxNM; have sPM := subset_trans (normG P) sNM. have{} sylP := pHall_subl sPM (subsetT M) sylP. by exists M => //; apply/exists_inP; exists P. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun eqtype ssrnat seq div path fintype bigop order) mathcomp(finset prime fingroup morphism perm automorphism quotient) mathcomp(action gproduct gfunctor pgroup cyclic center commutator) mathcomp(gseries nilpotent sylow abelian maximal hall frobenius) mathcomp(ssralg ssrnum ssrint rat) odd_order(BGsection1 BGsection3 BGsection4 BGsection5 BGsection6) odd_order(BGsection7 BGsection9 BGsection10 BGsection12 BGsection13)