# subnormal subgroup

Let $G$ be a group, and $H$ a subgroup of $G$. Then $H$ is a subnormal subgroup of $G$ if there is a natural number $n$ and subgroups $H_{0},\dots,H_{n}$ of $G$ such that

 $H=H_{0}\triangleleft H_{1}\triangleleft\cdots\triangleleft H_{n}=G,$

where $H_{i}$ is a normal subgroup of $H_{i+1}$ for $i=0,\dots,n-1$.

We may write $H\operatorname{sn}G$ or $H\triangleleft\triangleleft\,G$ or $H\!\trianglelefteq\trianglelefteq G$ to indicate that $H$ is a subnormal subgroup of $G$.

In a nilpotent group, all subgroups are subnormal.

Subnormal subgroups are ascendant and descendant.

 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