Burnside transfer theorem ($P\subseteq Z(N_G(P))⟹G\text{ has a normal }p\text{-complement}$)

The hypothesis makes $P$ abelian and makes any two conjugate elements of $P$ equal. For the latter assertion, Sylow conjugacy inside the centralizer of the target element adjusts a conjugating element into $N_G(P)$, where it centralizes $P$. Evaluating group transfer to $P$ on an element $u\in P$, each coset cycle contributes a conjugate of $u$ raised to the cycle length, hence that same power of $u$. The restriction is consequently $u\mapsto u^{[G:P]}$, an automorphism of the abelian $p$-group. Transfer is surjective, and its kernel is the required normal p-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.