$\pi$-separable group

Let $\pi$ be a set of prime numbers.A finite group $G$ is called $\pi$-separableif there exists a composition series

$$\{1\}=G_0⊲\mathrm{\cdots }⊲G_n=G$$

such that each $G_{i+1}/G_i$ is eithera $\pi$-group (http://planetmath.org/PiGroupsAndPiGroups)or a ${\pi }^{\prime }$-group (http://planetmath.org/PiGroupsAndPiGroups).

A $\{p\}$-separable group, where $p$ is a prime number,is usually called a $p$-separable group.

$\pi$-separability can be thought of asa generalization of solvability for finite groups;a finite group is $\pi$-separable for all sets of primesif and only it is solvable.