I want to check if my solution to this problem from my group theory course is correct:
Let $H,K\leq G$. We define $$[H,K]=\langle [h,k] : h\in H, k\in K\rangle.$$ Prove that $[H,K]=[K,H]$, and that $[G,H]\leq H \Longleftrightarrow H\unlhd G$.
My solution:
For the first prove, I've seen that, being $h\in H$ and $k\in K$, $$([h,k])^{-1}=(hkh^{-1}k^{-1})^{-1}=khk^{-1}h^{-1}=[k,h],$$ so this proves $[H,K]=[K,H]$.
For the second statement, I'll see the double implication:
$\boxed{\Rightarrow}$ Assume $[G,H]\leq H$. Then $[g,h]=ghg^{-1}h^{-1}\in H$, and given that $h^{-1}\in H$, we conclude $ghg^{-1}\in H$ so $H\unlhd G$.
$\boxed{\Leftarrow}$ Assume $H\unlhd G$. Then $\forall g\in G, h\in H$, $ghg^{-1}\in H$, and given that $h^{-1}\in H$, then $ghg^{-1}h^{-1}=[g,h]\in H$.
So from these we conclude that $[G,H]\leq H \Longleftrightarrow H\unlhd G$.
Is my solution correct? If not, why? Any help will be appreciated, thanks in advance.