$\pi$-separable group

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

$$\{1\}=G_0⊲\mathrm{\cdots }⊲G_n=G$$

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
Canonical name	piseparableGroup
Date of creation	2013-03-22 13:17:48
Last modified on	2013-03-22 13:17:48
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	7
Author	yark (2760)
Entry type	Definition
Classification	msc 20D10
Defines	$\pi$-separable
Defines	$p$-separable
Defines	$\pi$-separability
Defines	$p$-separability
Defines	$p$-separable group