Section PgroupProps.
Variable gT : finGroupType.
Implicit Types (pi rho : nat_pred) (p : nat).
Implicit Types (x y z : gT) (A B C D : {set gT}) (G H K P Q R : {group gT}).
(* These lemmas actually hold for maximal pi-groups, but below we'll *)
(* derive from the Cauchy lemma that a normal max pi-group is Hall.  *)

Lemma sub_normal_Hall pi G H K :
pi.-Hall(G) H -> H <| G -> K \subset G -> (K \subset H) = pi.-group K.