Prove that is nilpotent if and only if is a power of 2.
Suppose is nilpotent. Let be an odd prime dividing . Then is an element of order in ; in particular, . Now and are relatively prime, so that, by §6.1 #9, ; a contradiction. Thus no odd primes divide , and we have .
We proceed by induction on , where .
For the base case, , note that is abelian, hence nilpotent.
For the inductive step, suppose is nilpotent. Consider ; we have , and by §3.1 #34, is nilpotent. By §6.1 #6, is nilpotent.
Moreover, we show that for , is nilpotent of nilpotence class , by induction on .
For the base case, , is abelian, and thus of nilpotence class 1.
Suppose now that is nilpotent of nilpotence class . Since , by a previous lemma, is nilpotent of nilpotence class .