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.