Suppose $G$ is a group where $|G| = p^k$ where $p$ is a prime and $k\gt 0$. Prove that
- $|Z(G)| \gt 1$; and
- If $N$ is a normal subgroup of $G$ of order $p$, then $N$ is contained in $Z(G)$.
Suppose $G$ is a group where $|G| = p^k$ where $p$ is a prime and $k\gt 0$. Prove that