subnormal subgroup

Let G be a group, and H a subgroupMathworldPlanetmathPlanetmath of G. Then H is a subnormal subgroupMathworldPlanetmath of G if there is a natural numberMathworldPlanetmath n and subgroups H0,,Hn of G such that


where Hi is a normal subgroupMathworldPlanetmath of Hi+1 for i=0,,n-1.

We may write HsnG or HG or HG to indicate that H is a subnormal subgroup of G.

In a nilpotent groupMathworldPlanetmath, all subgroups are subnormal.

Subnormal subgroups are ascendant and descendantPlanetmathPlanetmath.

Title subnormal subgroup
Canonical name SubnormalSubgroup
Date of creation 2013-03-22 13:16:27
Last modified on 2013-03-22 13:16:27
Owner yark (2760)
Last modified by yark (2760)
Numerical id 21
Author yark (2760)
Entry type Definition
Classification msc 20D35
Classification msc 20E15
Synonym subinvariant subgroup
Synonym attainable subgroup
Related topic SubnormalSeries
Related topic ClassificationOfFiniteNilpotentGroups
Related topic NormalSubgroup
Related topic CharacteristicSubgroup
Related topic FullyInvariantSubgroup
Defines subnormal
Defines subnormality