The derived series removes the largest abelian quotient one layer at a time and measures how many abelian extensions are needed to build a group. Termination at the identity is precisely solvability. [@isaacs2008]
Definition
The least \(d\) with \(G^{(d)}=1\) is the derived length. With this convention a nontrivial abelian group has length \(1\).
Core result
Every factor \(G^{(i)}/G^{(i+1)}\) is abelian. The extension result uses \((G/N)^{(i)}=G^{(i)}N/N\): once the quotient series terminates, the derived series lies in \(N\) and then terminates there.
Structural properties
- Every \(G^{(i)}\) is characteristic in \(G\).
- Subgroups and quotients have derived length no larger than the original; solvable groups are closed under extensions.
- Nilpotent groups are solvable, but solvable groups such as \(S_3\) need not be nilpotent.
- A perfect group has \(G'=G\), so its derived series does not descend and it is nonsolvable unless trivial.
- For a finite group, the stable terminal subgroup \(G^{(\infty)}\) is the smallest normal subgroup with solvable quotient.
The derived series measures abelian factors, while the lower central series measures central factors. Both use commutators but encode different structure.
Example and boundary
Knowledge network
The derived subgroup supplies one step, the solvable-group article develops extension structure, and the lower central series provides the nilpotent comparison.
Proof and sources
Derived series, solvable extensions, and the stable terminal term are in Section 4B; the main proof follows from quotient compatibility of derived subgroups. [@isaacs2008]
