Solvable Group

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

• $G^{(0)}=G$, and
• $G^{(k+1)}=[G^{(k)},G^{(k)}]$ is the commutator subgroup of $G^{(k)}$.

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

Examples:

• Every abelian group is solvable (the derived series hits $\{e\}$ in at most two steps).
• $S_3$ is solvable.
• (Non-example) $A_5$ is not solvable.