Problem 5

Let $G$ be a finite group, and write the order of $G$ as $p^{k}m$ where $k\geq 0$, $m\geq 1$, and $p$ is a prime such that $p\nmid m$. Let $N\lhd G$ such that $p$ does not divide the index of $N$ in $G$. Prove that $N$ contains all Sylow $p$-subgroups of $G$ and vice versa. That is,

1. prove that any Sylow $p$-subgroup of $N$ is also in $G$; and

2. prove that any Sylow $p$-subgroup of $G$ is also in $N$.