End Pquotient.
(* Application of card_Aut_cyclic to internal faithful action on cyclic *)
(* p-subgroups.                                                         *)
Section InnerAutCyclicPgroup.
Variables (gT : finGroupType) (p : nat) (G C : {group gT}).
Hypothesis nCG : G \subset 'N(C).

(* Specifications: *)
Lemma logn_quotient_cent_cyclic_pgroup :
p.-group C -> cyclic C -> logn p #|G / 'C_G(C)| <= (logn p #|C|).-1.
Lemma p'group_quotient_cent_prime :
prime p -> #|C| %| p -> p^'.-group (G / 'C_G(C)).