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 $\mathrm{Aut}(P)$, 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.