---
tag: 096HN851YZ
kind: definition
name: Subnormal series
aliases: [subinvariant-series]
msc: ["20E15"]
defines: 0ZZR8D7Y1Z
uses: [0W77CVBCHQ, 0YYWCNTPWR]
sources:
  - url: https://en.wikipedia.org/wiki/Subgroup_series
    locator: Normal series, subnormal series
  - url: https://en.wikipedia.org/wiki/Subgroup_series
    locator: Definition
---

## Definition

Let $G$ be a [[group|G5S1M9]].
A subnormal series of $G$ is a finite sequence of subgroups $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$, where each $G_{i-1}$ is a [[normal subgroup|0CB0DX5BVV]] of $G_i$.
The [[quotient groups|0KYX2N69J0]] $G_i / G_{i-1}$ are the factors of the series, and $n$ is its length.

## Remarks

The subgroup $G_{i-1}$ needs to be normal only in $G_i$, not in $G$.
When each $G_i$ is normal in $G$, the series is called a normal series.
For example, $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$ is a normal series of $S_4$, where $V$ is the [[Klein four-group|0PMRQJVTME]] in $S_4$.

## Explanations

### L1 human

A subnormal series cuts a group into layers.
Each layer is a quotient group, and you can study the layers one at a time.
Good series have simple layers, like composition series, or abelian layers, like the series of a solvable group.