End MorphPcore.
Section EqPcore.
Variables gT : finGroupType.
Implicit Types (pi rho : nat_pred) (G H : {group gT}).

(* Specifications: *)
Lemma sub_in_pcore pi rho G :
{in \pi(G), {subset pi <= rho}} -> 'O_pi(G) \subset 'O_rho(G).
Lemma sub_pcore pi rho G : {subset pi <= rho} -> 'O_pi(G) \subset 'O_rho(G).
Lemma eq_pcore pi rho G : pi =i rho -> 'O_pi(G) = 'O_rho(G).
Lemma bigcap_p'core pi G :
G :&: \bigcap_(p < #|G|.+1 | (p : nat) \in pi) 'O_p^'(G) = 'O_pi^'(G).
Lemma coprime_pcoreC (rT : finGroupType) pi G (R : {group rT}) :
coprime #|'O_pi(G)| #|'O_pi^'(R)|.