Showing normal subgroup is a subgroup of the center.

Question

Given $N\lhd G$ and $N \cap G'=e$. Show that $N\leq C(G)$ where $G'$ is commutator group.

What exactly do we get from the hypothesis $N \cap G'=e$?

Answer 1

Since $N$ is normal, it is true that for all $g \in G$, $gNg^{-1} = N$. In particular, given $n \in N$, we have that

$$gng^{-1} \in N$$

Now there's a natural way to introduce a commutator, since we have

$$gng^{-1} n^{-1} \in N$$

What can you conclude from this?