Let $G$ be a finite group. Show that $G$ is not solvable iff it contains a normal non trivial subgroup $H$ with $H=H'$.
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I have done the following :
it holds that $H'=H^{(1)}=[H,H]$, derived group.
We assume that $G$ is solvable. Then each subgroup and quotient group of $G$ is solvable. So $H$ is solvable.
A group $H$ is solvable iff the derived series ends with the trivial group.
That cannot hold with a non trivial group $H$ with $H=H'$, because all derived groups $H^{(i+1)}=[H^{(i)}, H^{(i)}]$ are equal to $H$ which is non trivial.
So we get a contradiction.
Therefore $G$ cannot be solvable.
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Is everything correct and complete? Or isn't this enough for the iff statement?