Sylow number equals index of Sylow normalizer

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This article gives a proof/explanation of the equivalence of multiple definitions for the term Sylow number
View a complete list of pages giving proofs of equivalence of definitions

Statement

Let G be a finite group, p be a prime number, and P a p-Sylow subgroup (?) of G. Then, if n_p denotes the number of p-Sylow subgroups, we have:

[G:N_G(P)] = n_p

Facts used

  1. Sylow implies order-conjugate: Any two p-Sylow subgroups are conjugate.
  2. Group acts on set of subgroups by conjugation: Under this action, the isotropy subgroup for any subgroup is its normalizer, and the index of the normalizer equals the number of conjugate subgroups to it.

Proof

The proof follows directly by piecing together facts (1) and (2).