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  "body": "\n## Proof\n\nWe apply the orbit-stabiliser theorem to the action of $G$ by conjugation on its Sylow subgroups.\n\n1. Let $\\Omega = \\operatorname{Syl}_p(G)$ be the set of [[Sylow $p$-subgroups|0JWZX6FKSH]] of $G$, so $|\\Omega| = n_p$.\n2. For $g \\in G$ and $Q \\in \\Omega$, the [[conjugate subgroup|0D6WHTTH7T]] $gQg^{-1}$ has the same order as $Q$, because $x \\mapsto gxg^{-1}$ is a bijection. So $gQg^{-1} \\in \\Omega$.\n3. Thus $G$ acts on $\\Omega$ by $g \\cdot Q = gQg^{-1}$. This is the [[action by conjugation|0P8CY9V437]] restricted to $\\Omega$.\n4. By the [[second Sylow theorem|09GCQ5STVJ]], each $Q \\in \\Omega$ equals $gPg^{-1}$ for some $g \\in G$. So the [[orbit|00F2JFA41J]] $G \\cdot P$ is all of $\\Omega$.\n5. The [[stabiliser|0RGGF8ZS6D]] of $P$ is $\\{g \\in G : gPg^{-1} = P\\}$, which is the [[normaliser|0AT1ZSNM25]] $N_G(P)$.\n6. By the [[orbit-stabiliser theorem|0E2HSCKZ6C]], $|G \\cdot P| = [G : N_G(P)]$.\n7. By steps 4 and 6, $n_p = |\\Omega| = [G : N_G(P)]$.\n\n## Idea\n\nAll Sylow $p$-subgroups are conjugate, so they form one orbit under conjugation. The size of an orbit is the index of the stabiliser, and the stabiliser of $P$ is its normaliser.\n",
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