Solvable Groups

Introduction

The concept of a solvable group originated in Galois's work on the solvability of polynomial equations by radicals. Solvable groups are those that can be built from abelian groups through a series of extensions.

Definition

Definition 8.3: A group $G$ is solvable if its derived series terminates in the trivial subgroup, i.e., $G^{(n)} = {e}$ for some integer $n$ . An equivalent definition is that $G$ has a subnormal series with abelian factors.

Properties

The class of solvable groups is closed under taking subgroups, quotients, and extensions. That is, if $N ⊴ G$ , then $G$ is solvable if and only if both $N$ and $G / N$ are solvable.

Examples

Example 1: Abelian Groups

All abelian groups are solvable (their derived series terminates at the first step).

Example 2: Finite p-Groups

All finite $p$ -groups are solvable.

Example 3: Symmetric Groups

The symmetric group $S_{n}$ is solvable for $n \leq 4$ , but not for $n \geq 5$ .

The symmetric group $S_{5}$ is not solvable because its derived series is $S_{5} ⊵ A_{5} ⊵ A_{5} ⊵ \dots$ , which never reaches the trivial group since $A_{5}$ is simple and non-abelian (and thus a perfect group, with $[A_{5}, A_{5}] = A_{5}$ ).

Example 4: Dihedral Groups

The dihedral group $D_{n}$ is solvable for all $n$ . For example, in $D_{4}$ :

$D_{4}^{'} = ⟨ r^{2} ⟩$ (the subgroup generated by the rotation by 180°)
$D_{4}^{″} = {e}$

So the derived series is $D_{4} ⊵ ⟨ r^{2} ⟩ ⊵ {e}$ , showing that $D_{4}$ is solvable.

Connection to Galois Theory

The connection to Galois theory is profound: a polynomial equation is solvable by radicals if and only if its Galois group is a solvable group.

This is why there is no general formula for solving quintic equations by radicals—the Galois group of a general quintic is $S_{5}$ , which is not solvable.

Applications

Application 1: Galois Theory

The concept of solvable groups is fundamental to Galois theory. A polynomial equation is solvable by radicals if and only if its Galois group is solvable.

Application 2: Group Classification

Solvable groups are important in the classification of finite groups. Many important families of groups are solvable.

Application 3: Structure Analysis

Understanding solvable groups helps us understand the structure of more complex groups.

Metabelian Groups

A group $G$ is metabelian if $G^{″} = {e}$ (i.e., the second derived subgroup is trivial). Metabelian groups are solvable groups that are "close" to being abelian. Examples include:

All abelian groups
The dihedral groups $D_{n}$
The symmetric groups $S_{n}$ for $n \leq 4$