---
tag: FKTQ1WBMKM
kind: concept
name: Subnormal subgroup
msc: [20D35, "20E15"]
specializes: [0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Subnormal_subgroup
same_as:
  wikidata: Q2361983
  wikipedia: Subnormal subgroup
---

## Summary

A subnormal subgroup is joined to the whole group by a finite chain of subgroups, each normal in the next.
Every normal subgroup is subnormal, but not conversely, because normality is not transitive.
Subnormal subgroups are exactly the subgroups that occur as terms of subnormal series, and they are central in the structure theory of finite groups.

## Explanations

### L1 human

Being normal is a relation between a subgroup and a bigger group.
If $H$ is normal in $K$ and $K$ is normal in $G$, then $H$ need not be normal in $G$.
A subnormal subgroup is one that reaches the whole group in finitely many such normal steps.
For example, in the alternating group $A_4$ the subgroup $\{e, (1\,2)(3\,4)\}$ is normal in the Klein four-group, which is normal in $A_4$.
So it is subnormal in $A_4$, even though it is not normal in $A_4$.