Equivalence of definitions of p-constrained group

Statement
The following are equivalent for a finite group $$G$$ and a prime number $$p$$:


 * 1) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, we have $$C_G(P \cap O_{p',p}(G)) \le O_{p',p}(G)$$.
 * 2) Let $$H = G/O_{p'}(G)$$. Then, the p-core of $$H$$ is a self-centralizing subgroup of $$H$$, i.e., $$C_H(O_p(H)) \le H$$.