Equivalence of definitions of p-constrained group

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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)) \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.

Proof

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