# $\pi$-separable group

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

 $\{1\}=G_{0}\lhd\cdots\lhd G_{n}=G$

such that each $G_{i+1}/G_{i}$ is either a $\pi$-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 as a generalization of solvability for finite groups; a finite group is $\pi$-separable for all sets of primes if and only it is solvable.

Title $\pi$-separable group piseparableGroup 2013-03-22 13:17:48 2013-03-22 13:17:48 yark (2760) yark (2760) 7 yark (2760) Definition msc 20D10 $\pi$-separable $p$-separable $\pi$-separability $p$-separability $p$-separable group