# Equivalence of definitions of p-constrained group

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., $C_H(O_p(H)) \le H$.