If we consider a finite group G, a prime p, and we suppose that the Sylow p-subgroup P is normal in G, we know by Schur-Zassenhaus that $G\cong P\rtimes G/P$.
Is there an elementary way to see it? In particular, I'd like to identify the complement subgroup in $G$ which is isomorphic to $G/P$. I was pretty sure that this was just an easy check if we use Sylow Thms, but I haven't managed to see it so far. Thanks!