Derived length is logarithmically bounded by nilpotency class
- Nilpotent implies solvable
- Derived length gives no upper bound on nilpotency class: Knowing the derived length of a nilpotent group gives no bound on its nilpotence class. In fact, there are metabelian groups of arbitrarily large class, such as the dihedral -groups and the generalized quaternion groups.
- Second half of lower central series of nilpotent group comprises abelian groups: If is nilpotent of class , and denotes the term of the lower central series of , then is abelian for .
We prove this by induction on the nilpotence class. Note that the statement is true when or .
Given: A finite nilpotent group of class .
To prove: The derived length of is at most .
Proof: Let be the smallest positive integer greater than or equal to . In other words, either or , depending on the parity of . Then, is an abelian group, and is a group of class , which is at most .
By the induction assumption, we have:
Thus, has an abelian normal subgroup such that the derived length of the quotient is at most . This yields that the derived length of is at most .