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.