# $\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

^{}

$$\{1\}={G}_{0}\u22b2\mathrm{\cdots}\u22b2{G}_{n}=G$$ |

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 |