Derived series

Let $G$ be a group . The derived series of $G$ is the sequence of subgroups $(G^{(n)})_{n\ge 0}$ defined recursively by $G^{(0)}=G,\qquad G^{(n+1)}=[G^{(n)},G^{(n)}],$ where $[G^{(n)},G^{(n)}]$ denotes the commutator subgroup of $G^{(n)}$. Each $G^{(n)}$ is a normal subgroup of $G$, and the quotients $G^{(n)}/G^{(n+1)}$ are abelian.

A group $G$ is solvable if there exists $n$ such that $G^{(n)}$ is the trivial subgroup . The derived series is a particular kind of subnormal series that measures how far $G$ is from being abelian.

Examples:

• If $G$ is abelian, then $G^{(1)}=\{e\}$, so the derived series terminates after one step.
• For $S_3$, one has $S_3^{(1)}=A_3$ and $S_3^{(2)}=\{e\}$, so $S_3$ is solvable of derived length $2$.
• For a nonabelian simple group (e.g. $A_5$), the derived series does not reach $\{e\}$, hence such a group is not solvable.