Let $G = H_1 \times H_2$. Let $N$ be a normal subgroup of $G$ such that $N \cap H_1 = N\cap H_2 = \{e\}$. Prove that $N$ lies in the center of $G$. I can see that $G$ is a direct product of $H_1$ and $H_2$. How can $N \cap H_1 = N \cap H_2 = \{e\}$ lead to the conclusion that $N$ lies in the center? Do I need to use the fact that the center of a group is a normal subgroup (which would be $N$ here)?  I feel like I can't use that assumption to prove this...
Any insight will be greatly appreciated!

Note that $H_1$ and $H_2$ are normal subgroups in G. Thus $N\cap H_1=\{e\}$ implies $H_1$ centralizes $N$,hence G centralizes N. So N lies in the center of G.
