= Burnside transfer theorem
{c}
{title2=$P\subseteq Z(N_G(P))\Longrightarrow G\text{ has a normal }p\text{-complement}$}
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>.
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