proposition: Lemma p_rank_quotient G H : G \subset 'N(H) -> 'r_p(G) - 'r_p(H) <= 'r_p(G / H).
proof: Proof. move=> nHG; rewrite leq_subLR. have [E EpE] := p_rank_witness p G; have{EpE} [sEG abelE <-] := pnElemP EpE. rewrite -(LagrangeI E H) lognM ?cardG_gt0 //. rewrite -card_quotient ?(subset_trans sEG) // leq_add ?logn_le_p_rank // !inE. by rewrite subsetIr (abelemS (subsetIl E H)). by rewrite quotientS ?quotient_abelem. Qed.
imports: mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path choice) mathcomp(div fintype finfun bigop finset prime binomial) mathcomp(fingroup morphism perm automorphism action) mathcomp(quotient gfunctor gproduct ssralg countalg finalg zmodp) mathcomp(cyclic pgroup gseries nilpotent sylow)