---
tag: 0P3KZK0FTV
kind: definition
name: Solvable group
msc: [20F16]
defines: 0FXJ7YFMMX
uses: [096HN851YZ, 071DDFVG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Solvable_group
    locator: Definition
---

## Definition

A [[group|G5S1M9]] $G$ is solvable when it has a [[subnormal series|0ZZR8D7Y1Z]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is [[abelian|082GHNPJ53]].

## Remarks

Equivalently, the [[derived series|0Q5G5DJ4P2]] of $G$ reaches $\{e\}$ after finitely many steps.
A finite group is solvable if and only if its [[composition factors|0PHTC93M58]] are all cyclic of prime order.
The group $S_4$ is solvable, by the series $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$.
The name comes from Galois theory: a polynomial equation over $\mathbb{Q}$ is solvable by radicals if and only if its Galois group is solvable.

## Explanations

### L1 human

A solvable group can be built from abelian groups in finitely many layers.
Abelian groups are easy to understand, so solvable groups are close to easy.
The smallest group that is not solvable is $A_5$, of order $60$.