Derived series - Imported Knowlpedia Content

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.

Remarks

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.