---
tag: AWWM9RWETX
kind: concept
name: Supersolvable group
msc: [20F16, 20D10]
specializes: [0FXJ7YFMMX]
sources:
  - url: https://en.wikipedia.org/wiki/Supersolvable_group
same_as:
  wikidata: Q7644153
  wikipedia: Supersolvable group
---

## Summary

A supersolvable group is a group with a finite chain of normal subgroups, all normal in the whole group, whose successive quotients are cyclic.
The condition is stronger than solvability and weaker than nilpotency for finite groups.
Supersolvable groups share many good properties of nilpotent groups, and they appear naturally in the study of finite solvable groups and their representations.

## Explanations

### L1 human

A solvable group can be built from abelian layers.
A supersolvable group can be built from cyclic layers, and each partial stack of layers is normal in the whole group.
The symmetric group $S_3$ is supersolvable: it has the normal subgroup $A_3$, which is cyclic of order $3$, and the quotient $S_3/A_3$ is cyclic of order $2$.
The alternating group $A_4$ is solvable but not supersolvable.