This is a article which Antonio Beltran. I'm reading lemma 2.2.b). I see that:
"Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every prime p,
(b) $ν_p(C)$ divides $ν_p^A (G)$ and $ν_p^A (G)$ divides $ν_p(G)$.
Proof:
(b) If P is an A-invariant Sylow subgroup of G, then by coprime action properties $P ∩C$ is a Sylow p-subgroup of C and obviously, $N_C (P) ⊆ N_C (P ∩C)$. Hence $ν_p(C) = |C : N_C (P ∩ C)|$ divides $ν_p^A (G) = |C : N_C (P)|$. On the other hand, by applying Lemma 2.1 to the A-invariant subgroup $N_G(P)$, we get that $ν_p^A (G) = |C : N_C (P)|$ divides $ν_p(G) = |G : N_G(P)|$."
I don't understand why "if $N_C (P) ⊆ N_C (P ∩C)$, then $ν_p(C) = |C : N_C (P ∩ C)|$ divides $ν_p^A (G) = |C : N_C (P)|$."
Thank you very much.