# 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