Prove that if is a nontrivial subgroup of a solvable group then there is a nontrivial subgroup which is abelian and normal in .
We proceed by induction on the width of – that is, the number of prime factors of (including multiplicity).
If has width 1, then is simple and no such exists.
If has width 2, say , and is nontrivial, then without loss of generality .
Lemma: Let be a nonempty partially ordered set such that every chain has bounded length. Then there exists a minimal element . Proof: Let be a chain of maximal length , where . If an element exists such that , then is a chain of length , a contradiction. So is -minimal in .
Consider the set . Note that is nonempty since . Note that because is solvable, has a finite composition series, so that by the Jordan-Holder Theorem, all composition series for have the same finite length . Now every chain in can be extended to a composition series for , and thus has length bounded by . By the lemma, then, there exists a -minimal element . By Lemma 5 to this previous theorem, is abelian.