Equivalence of definitions of p-constrained group

This article gives a proof/explanation of the equivalence of multiple definitions for the term p-constrained group
View a complete list of pages giving proofs of equivalence of definitions

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 $C_{G}(P\cap O_{p',p}(G))\leq O_{p',p}(G)$.
2. Let $H=G/O_{p'}(G)$. Then, the p-core of $H$ is a self-centralizing subgroup of $H$, i.e., $C_{H}(O_{p}(H))\leq H$.

Proof

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