Problem 5.4.16 (If $K \unlhd G$ then $K' \unlhd G$.)

Question

Prove that if $K$ is a normal subgroup of $G$ then $K' \unlhd G$.

Richard Ganaye · Accepted Answer

\begin{proof} 
If $h,k \in K$, then for all $g \in G$, $ghg^{-1} \in K$ and $gk g^{-1} \in K$, thus
$$g[h,k] g^{-1} = [ghg^{-1}, g k g^{-1}] \in K'.$$
Since $K'$ is generated by the commutators $[h,k], \ (h, k \in K)$, then $K' \unlhd G$.

\end{proof}

Problem 5.4.16 (If $K \unlhd G$ then $K' \unlhd G$.)

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Problem 5.4.16 (If $K \unlhd G$ then $K' \unlhd G$.)

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