# The center of the quotient group is the quotient group of the center

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..

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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).
@PhilipBenjMarcobyEragon We can write "$zN\in Z(G/N)$" as $\forall g\in G,[zN,gN]=N$, and write the statement "$z\in Z(G)$" as $\forall g\in G,[z,g]=1$. Note that $[z,g]\in[G,G]$ is automatic. In these sorts of elementary problems, to get an intuitive grip on the meanings behind these things, you'll want to practice and experiment with writing out statements in as many equivalent ways as possible. Note that $Z(G/N)=Z(G)/N$ may be written as $zN\in Z(G/N)\iff z\in Z(G)$. – anon Mar 19 '13 at 6:35