Prove that $N \setminus Z(G)\neq \langle e \rangle$.

Question

Let $G$ be a group with $\operatorname{ord}(G) = p^n$, where $p$ is a prime number, and if $N \neq \langle e \rangle$ is a normal subgroup of $G$, prove that $N \setminus Z(G)\neq \langle e \rangle$.

Answer 1

I think you might mean $N\cap Z(G) \neq \{e\}$. In which case,

1. Use the Class equation
2. If $x\in N$, then the conjugacy class $C(x) \subset N$