---
tag: 7TP5M906HB
kind: definition
name: Supersolvable group
msc: [20F16, 20D10]
defines: AWWM9RWETX
uses: [0CB0DX5BVV, 0KYX2N69J0, 0KVGH0J5B3]
sources:
  - url: https://en.wikipedia.org/wiki/Supersolvable_group
---

## Definition

Let $G$ be a [[group|G5S1M9]].
The group $G$ is *supersolvable* when it has a chain of subgroups $\{e\} = G_0 \le G_1 \le \cdots \le G_n = G$ with this property.
Each $G_i$ is a [[normal subgroup|0CB0DX5BVV]] of $G$ and each [[quotient group|0KYX2N69J0]] $G_i/G_{i-1}$ is [[cyclic|0KVGH0J5B3]].

## Remarks

Example: $S_3$ is supersolvable with the series $\{e\} \le A_3 \le S_3$.
Here $A_3$ and $S_3$ are normal in $S_3$, $A_3 \cong \Z/3\Z$ and $S_3/A_3 \cong \Z/2\Z$.
Non-example: $A_4$ is not supersolvable.
Its only normal subgroups are $\{e\}$, the Klein four-group $V$ and $A_4$.
So the first term of such a series that differs from $\{e\}$ is $V$ or $A_4$, and the factor at that step is $V$ or $A_4$, which is not cyclic.
Every finitely generated [[nilpotent group|0Q7MMQHC1P]] is supersolvable, and every supersolvable group is [[polycyclic|9RBA56NXX3]], hence [[solvable|0FXJ7YFMMX]].
Some authors write supersoluble.

## Explanations

### L1 human

A group is supersolvable when it can be built from cyclic layers, with every partial stack normal in the whole group.