proposition: Lemma sigma'_rank2_beta' : 'r_p(M) = 2 -> p \notin \beta(G).
proof: Proof. move=> rpM; rewrite -[p \in _]negb_exists_in negbK; apply/exists_inP. have [A Ep2A]: exists A, A \in 'E_p^2(M) by apply/p_rank_geP; rewrite rpM. have [_ abelA dimA] := pnElemP Ep2A; have [pA _] := andP abelA. have [P sylP sAP] := Sylow_superset (subsetT _) pA. exists P; rewrite ?inE //; apply/implyP=> _; apply/set0Pn. exists A; rewrite 3!inE abelA dimA sAP (subsetP (pmaxElemS _ (subsetT P))) //. by rewrite inE (subsetP (sigma'_rank2_max rpM)) // inE. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun eqtype ssrnat seq div path fintype) mathcomp(bigop finset prime fingroup morphism perm automorphism quotient) mathcomp(action gproduct gfunctor pgroup cyclic center commutator) mathcomp(gseries nilpotent sylow abelian maximal hall) odd_order(BGsection1 BGsection3 BGsection4 BGsection5 BGsection6) odd_order(BGsection7 BGsection9)