Solvable Group - Knowlpedia

A solvable group is a group $G$ whose derived series reaches the trivial subgroup after finitely many steps. Explicitly, there is an integer $n\ge 0$ such that $G^{(n)}=\{e\}$ , where

G^{(0)}=G, \qquad G^{(k+1)}=[G^{(k)},G^{(k)}].

Here $[G^{(k)},G^{(k)}]$ is the commutator subgroup of $G^{(k)}$ .

Examples

Every abelian group is solvable: $G^{(1)}=\{e\}$ .
$S_3$ is solvable.
(Non-example) $A_5$ is not solvable.

Equivalent characterizations

Equivalently, $G$ is solvable iff it has a finite subnormal series whose successive quotients are abelian.