Prove that if $G$ is a finite nilpotent group and $N\trianglelefteq G$, then $N\cap Z(G)\ne\{e\}$.
This is true for all $p$-groups.
Suppose $|G|=p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ where $p_1,\cdots,p_s$ are distinct primes, then $G\cong P_1\times\cdots\times P_s$ where $P_i$ is a Sylow $p_i$-subgroup.
Let $\varphi$ be an isomorphism from $G$ to $P_1\times\cdots\times P_s$. Since $Z(G)\cong Z(P_1)\times\cdots\times Z(P_s)$, I want to prove that $\varphi(N)\cap[Z(P_1)\times\cdots\times Z(P_s)]\ne\{e\}$, then the inequality is hold. But I don't know how to do this.