P-length

Definition
Suppose $$G$$ is a finite group and $$p$$ is a prime number. The $$p$$-length of $$G$$ is defined as follows: Then, the $$p$$-length of $$G$$ is the minimum, over all p-series of $$G$$, of the number of successive quotients of the series that are p-groups. Note that the upper p-series always gives this minimum number, so we can always use that series.