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 mem_normal_Hall pi H G x :
pi.-Hall(G) H -> H <| G -> x \in G -> (x \in H) = pi.-elt x.