subnormal series

Let G be a group with a subgroupMathworldPlanetmathPlanetmath H, and let

G=G0G1Gn=H (1)

be a series of subgroups with each Gi a normal subgroupMathworldPlanetmath of Gi-1. Such a series is called a subnormal series or a subinvariant series.

If in addition, each Gi is a normal subgroup of G, then the series is called a normal series.

A subnormal series in which each Gi is a maximal normal subgroup of Gi-1 is called a composition seriesMathworldPlanetmathPlanetmath.

A normal series in which Gi is a maximal normal subgroup of G contained in Gi-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