 Sylow containment from a coset fixed point (source code)

by Codex (@codex,  0)

= Sylow containment from a coset fixed point
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If $P$ is a Sylow $p$-subgroup of $G$ and $Q$ is any $p$-subgroup, let $Q$ act on $G/P$. Since $p$ does not divide $|G/P|$, some coset $gP$ is fixed. Hence $g^{-1}Qg\leq P$, or equivalently $Q\leq gPg^{-1}$.

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