Derived series

The derived series of G is the chain of iterated commutator subgroups. It measures how many times we have to take commutators before G collapses to the identity — and G is solvable exactly when this collapse happens.

Definition

The derived (or commutator) subgroup is G' = [G, G], generated by all commutators [a, b] = a⁻¹b⁻¹ab. Iterating: set G⁽⁰⁾ = G and

G⁽ⁿ⁺¹⁾ = [G⁽ⁿ⁾, G⁽ⁿ⁾].

The chain

G = G⁽⁰⁾ ⊇ G⁽¹⁾ ⊇ G⁽²⁾ ⊇ ⋯

is the derived series of G. Each G⁽ⁿ⁾ is a characteristic (hence normal) subgroup of G, and G / G' is the abelianization — the largest abelian quotient of G.

G is perfect if G⁽¹⁾ = G, i.e. [G, G] = G; equivalently, if G has no non-trivial abelian quotient.

Properties

Monotone. G⁽ⁿ⁺¹⁾ ⊆ G⁽ⁿ⁾.
Quotients are abelian. Each G⁽ⁿ⁾ / G⁽ⁿ⁺¹⁾ is abelian by construction.
Stabilizes in finite groups. There is a smallest n at which G⁽ⁿ⁾ = G⁽ⁿ⁺¹⁾; this common term is the largest perfect subgroup of G, sometimes called the perfect core G^(∞).
Finite groups split into two cases. Either G⁽ⁿ⁾ = {e} for some n (then G is solvable) or G⁽∞⁾ ≠ {e} (then G contains a non-trivial perfect subgroup).

Derived length

If the derived series reaches {e}, the smallest n with G⁽ⁿ⁾ = {e} is the derived length (or solvable length) of G.

Derived length 0: G = {e}.
Derived length 1: G is non-trivial abelian.
Derived length 2: G is metabelian — has abelian [G, G] but is not abelian.

Examples

Abelian G. G⁽¹⁾ = {e}. Derived length ≤ 1.
S₃ — S₃⁽¹⁾ = A₃ = C₃, S₃⁽²⁾ = {e}. Derived length 2.
S₄ — S₄ ⊳ A₄ ⊳ V₄ ⊳ {e} (the last step is V₄⁽¹⁾ = {e}). Derived length 3.
A₅ — A₅⁽¹⁾ = A₅ (it is perfect). The derived series stabilizes at A₅; A₅ is not solvable.
Sₙ for n ≥ 5 — perfect core is Aₙ.

Relation to the lower central series

The lower central series γ₁ = G ⊇ γ₂ ⊇ ⋯ and the derived series are related by G⁽ⁿ⁾ ⊆ γ_n+1. In particular:

Nilpotent (lower central reaches {e}) ⟹ solvable (derived reaches {e}).
The converse fails: S₃ is solvable but not nilpotent.

Computation

Polycyclic group. G⁽¹⁾ is generated by pc-commutators of generators; iterate. Each step is polynomial in the number of generators.
Arbitrary finite group. Iterate H ↦ [H, H] using normal closure; stop when H stops shrinking.