= Cyclic Sylow subgroup at the least prime gives a normal complement
Let $p$ be the smallest prime dividing the order of a <finite group>, and suppose its <Sylow subgroup> $P$ is cyclic of order $p^a$. <Conjugation> embeds $N_G(P)/C_G(P)$ into $\operatorname{Aut}(P)$, of order $p^{a-1}(p-1)$. Since $P\subseteq C_G(P)$, this quotient has order prime to $p$. Its remaining possible prime divisors are smaller than $p$ and cannot divide $|G|$. Thus $N_G(P)=C_G(P)$, and the <Burnside transfer theorem> gives a <normal p-complement>.
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