subnormal series

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

G=G_{0}\rhd G_{1}\rhd\cdots\rhd 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