Show that if $G$ is a finite nonsolvable group then $G$ contains a nontrivial subgroup $H$ such that $\left[H,H\right]=H$.
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Note that $[H,H]\leqslant H$ for any group $H$. Thus, by contrapositive, if $G$ contains no non-trivial subgroup $H$ so that $[H,H]=H$, then $[H,H]$ is a proper subgroup of $H$ for each nontrivial subgroup $H$ of $G$. In particular, every term of the derived series of $G$ is strictly smaller than the previous term, whence the series converges to $1$, so $G$ is solvable.
Hint: If $\,G\,$ is simple (and non-abelian, of course) the claim is trivial (why?), so we can assume $\,G\,$ has a non-trivial normal subgroup. Now take a peek at a non-trivial normal minimal subgroup.