Problem 5.4.2 ($H \unlhd G \iff [G,H] \leq H$)

Question

Prove that a subgroup $H$ of $G$ is normal if and only if $[G,H] \leq H$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $H$ be a subgroup of $G$.

By Proposition 7, part (2),
$$H \unlhd G \iff [H,G] \leq H,$$
and by Exercise 1, $[H,G] = [G,H]$. Therefore
$$H \unlhd G \iff [G,H] \leq H.$$

\end{proof}

Problem 5.4.2 ($H \unlhd G \iff [G,H] \leq H$)

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Problem 5.4.2 ($H \unlhd G \iff [G,H] \leq H$)

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