Pi-separable group

Definition
Let $$G$$ be a finite group and $$\pi$$ be a set of primes (we may throw out all the members of $$\pi$$ that are not divisors of the order of $$G$$ -- these have no effect). We have the following equivalent formulations for saying that $$G$$ is a $$\pi$$-separable group:

$$G$$ is $$\pi$$-separable if and only if it is $$\pi'$$-separable, where $$\pi'$$ is the complement of $$\pi$$ in the set of prime divisors of the order of $$G$$.

The pi-length of $$G$$ is defined as the half-length of the lower pi-series, i.e., the number of successive quotients of the lower pi-series that are $$\pi$$-groups.