Let $P \in{\rm Syl}_p(G)$ and suppose $P$ is metacyclic. Suppose $(|G|, p^2 - 1) = 1$. Show that $G$ has a normal $p$ complement.
Since $P$ is metacycilc, let $N \unlhd P$ so that $P/N$ is cyclic. Let $m, k \in \mathbb{Z}$, such that $|P| = p^m$, and $|N| = p^k$ First we will show $N_{G/N}(P/N) = C_{G/N}(P/N)$. We know $$|N_{G/N}(P/N)/C_{G/N}(P/N)| \, / \, |\textbf{Aut}(P/N)| = \phi(p^{m - k}) = (p - 1)p^{m - k - 1}. $$ But $|$Aut$(P/N)| \, / \, |G|$, and $(|G|, (p - 1)(p + 1)) = (|G|, p^2 - 1) = 1$. So $$(|\textbf{Aut}(P/N)|, p - 1) = 1.$$ Therefore $$|N_{G/N}(P/N)/C_{G/N}(P/N)| \, / \, p^{m - k - 1}.$$ Now $$|N_{G/N}(P/N)/C_{G/N}(P/N)| \, / \, |(G/N)/C_{G/N}(P/N)| \, / \, |(G/N)/(P/N)|$$ , but since $(|G/N|, p) = 1$, we have $$(|N_{G/N}(P/N)/N_{G/N}(P/N)|, p) = 1.$$ Therefore $|N_{G/N}(P/N)/C_{G/N}(P/N)| = 1$, hence $N_{G/N}(P/N) = C_{G/N}(P/N)$. \newline \newline Now $N_{N_G(N)/N}(P/N) = (N_G(P)/N) \cap N_{G/N}(P/N)= (N_G(N)/N) \cap C_{G/N}(P/N)$ \newline $= C_{N_G(N)/N}(P/N)$. So by the correspondence theorem, $$N_{N_G}(N)(P) = C_{N_G}(N)(P)$$
Now $$|N_{G/N}(P/N)/C_{G/N}(P/N)| \, / \, |(G/N)/C_{G/N}(P/N)| \, / \, |(G/N)/(P/N)|$$ , but since $(|G/N|, p) = 1$, we have $$(|N_{G/N}(P/N)/N_{G/N}(P/N)|, p) = 1.$$ Therefore $|N_{G/N}(P/N)/C_{G/N}(P/N)| = 1$, hence $N_{G/N}(P/N) = C_{G/N}(P/N)$.
***My problem is I want to some how use $N_{G/N}(P/N) = C_{G/N}(P/N)$ to get $N_G(P) = C_G(P)$, but can't just use the correspondence theorem, since $N$ might not be normal in $G$
I can show $N_{N_G(N)/N}(P/N) = (N_G(P)/N) \cap N_{G/N}(P/N)= (N_G(N)/N) \cap C_{G/N}(P/N)$ $= C_{N_G(N)/N}(P/N)$. So by the correspondence theorem, $$N_{N_G(N)}(P) = C_{N_G(N)}(P)$$ and get $N_G(N)$ has a normal $p$ complement.
I feel like I'm some how supposed to use $p \in{\rm Syl}_p(G)$ or $P/N$ is cyclic, but not sure how?
***Edit: Looking at Frobenius's normal p-complement.
If $N \leq A \leq P$, then I can show $N_G(A)$ has a normal $p$-compliment, similarly to how I originally showed $N_G(N)$ has a normal p-complement, since $A \unlhd P$ and $P/A$ is cyclic. But not sure what to do if $N \nleq A$?