Aluffi's Algebra claims (implicitly) that any finite simple non-abelian group is not solvable. On the other hand, it defines a group to be not solvable if its derived series doesn't terminate with the identity. How to show that a finite simple non-abelian group is not solvable based on the original definition it gives?
PS The derived series of $G$ is the sequence of subgroups $G ⊇ G' ⊇ G'' ⊇ G''' ⊇ \dots$ . $G' = [G,G]$.
Well, we know that always $\;G'\lhd G\;$ , so if the group $\;G\;$ is non-abelian simple...what is then $\;G'\;$ ?