Pi-series

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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.