$G'$ is the smallest normal subgroup of $G=\langle K\rangle$ that contains $[K,K]$.

Question

I am a bit stuck with this problem. It seems like a sensible statement but I cannot prove it. What I have to prove is that if $G$ is generated by a generating set $K$ then $G' = [G,G]$ is the smallest normal subgroup of $G$ that contains $[K,K]$. I can see that the smallest normal subgroup of $G$ that contains $K$ is $\langle K \rangle$, i.e. $G$ itself. However, I am not sure this implies directly the result I am looking for.

Thanks in advance.

Answer 1

Let $N \unlhd G$ with $[K,K] \le N$. Then the images of any pair of elements of $K$ in $G/N$ commute with each other. But $G/N$ is generated by these images, so $G/N$ must be abelian, and hence $G' \le N$.