Lower central series is fastest descending central series
Then, if we consider the Lower central series (?) of :
Then, for every , we have:
In particular, if has Nilpotence class (?) , then:
We use the following fact:
- If and , then .
Consider the lower central series of :
To prove: For every , we have:
In particular, if has nilpotence class , then:
Proof: We prove this by induction on .
Base case for induction: For , so it's true.
Induction step: Suppose . Then by the definition of the lower central series:
Because , we have, by the fact mentioned above:
Finally, by the definition of central series, we have:
Combining these facts, we see that:
Thus, if is trivial, must be trivial. Since the smallest for which is trivial is , we have .