Pi-series

Definition
Let $$G$$ be a finite group and $$\pi$$ be a set of prime numbers. A $$\pi$$-series of $$G$$ is a subnormal series with the property that every factor group of the series is either a $$\pi$$-group (all the prime factors of its order are in $$\pi$$) or a $$\pi'$$-group (none of the prime factors of its order are in $$\pi$$).

If $$G$$ possesses a $$\pi$$-series, then $$G$$ is termed a pi-separable group. Two pi-series of particular interest are the upper pi-series and lower pi-series.