# subnormal series

Let $G$ be a group with a subgroup^{} $H$, and let

$$G={G}_{0}\u22b3{G}_{1}\u22b3\mathrm{\cdots}\u22b3{G}_{n}=H$$ | (1) |

be a series of subgroups with each ${G}_{i}$ a normal subgroup^{} of ${G}_{i-1}$.
Such a series is called a *subnormal series* or a *subinvariant series*.

If in addition, each ${G}_{i}$ is a normal subgroup of $G$,
then the series is called a *normal series*.

A subnormal series in which each ${G}_{i}$ is a maximal normal subgroup
of ${G}_{i-1}$ is called a *composition series ^{}*.

A normal series in which ${G}_{i}$ is a maximal normal subgroup of $G$ contained in ${G}_{i-1}$
is called a *principal series* or a *chief series*.

Note that a composition series need not end in the trivial group $1$.
One speaks of a composition series (1) as a *composition series from $G$ to $H$*.
But the term *composition series for $G$*
generally means a composition series from $G$ to $1$.

Similar remarks apply to principal series.

Some authors use normal series as a synonym for subnormal series. This usage is, of course, not compatible with the stronger definition of normal series given above.

Title | subnormal series |

Canonical name | SubnormalSeries |

Date of creation | 2013-03-22 13:58:42 |

Last modified on | 2013-03-22 13:58:42 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 8 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20D30 |

Synonym | subinvariant series |

Related topic | SubnormalSubgroup |

Related topic | JordanHolderDecompositionTheorem |

Related topic | Solvable |

Related topic | DescendingSeries |

Related topic | AscendingSeries |

Defines | composition series |

Defines | normal series |

Defines | principal series |

Defines | chief series |