[Math] The center of the quotient group is the quotient group of the center

Question

Let $N$ be a normal subgroup of a group $G$ such that $N\cap G'=\{e\}$, where $G'$ is the derived/commutator subgroup of $G$.
Then
i.) $N\subseteq Z(G)$, where $Z(G)$ is the center of $G$
ii.) $Z(G/N)=Z(G)/N$

Help me figure this out please…
Thank you..

Answer 1

For (1), show that $[G,N]$ is a subset of both $N$ (using its normality) and $[G,G]$.

And for (2), use $[zN,gN]=N\iff [z,g]N=N\iff [z,g]\in N$ (put $\forall g\in G$ in front if it helps).