Solvable groups

A finite group is solvable when it can be built from abelian pieces: it has a subnormal series whose successive factors are all abelian. Equivalently, its derived series reaches the trivial subgroup. The smallest non-solvable group is the alternating group A₅ of order 60.

Definition

For a finite group G, the following are equivalent.

• Subnormal series with abelian factors — there is a chain

  1 = G₀ ◁ G₁ ◁ ⋯ ◁ Gₙ = G,     each quotient Gᵢ₊₁/Gᵢ abelian.
  

• Derived series terminates — the iterated derived subgroup

  G = G⁽⁰⁾ ⊵ G⁽¹⁾ ⊵ G⁽²⁾ ⊵ ⋯,     G⁽ᵏ⁺¹⁾ = [G⁽ᵏ⁾, G⁽ᵏ⁾],
  

  reaches G⁽ⁿ⁾ = {e} for some n. Each G⁽ᵏ⁾ is characteristic in G. The least such n is the derived length of G.

• Prime-order composition factors — every composition factor of G is cyclic of prime order.

The derived series is the fastest-descending series with abelian factors, so it certifies solvability whenever any abelian-factor series exists.

Properties

• Closed under subgroups, quotients, and extensions. If G is solvable so is every subgroup and every quotient G/N. Conversely, if N ◁ G with both N and G/N solvable, then G is solvable. Hence a finite direct product of solvable groups is solvable. (These make the solvable groups a formation closed under extension.)
• No non-abelian simple factor. G is solvable iff none of its composition factors is a non-abelian simple group — the obstruction to solvability is exactly a non-abelian simple factor.
• Burnside's pᵃqᵇ theorem. Every finite group whose order is divisible by at most two distinct primes is solvable. See Burnside's theorem.
• Feit–Thompson theorem. Every finite group of odd order is solvable. Equivalently, every non-abelian finite simple group has even order.
• Philip Hall's theorem. G is solvable iff for every set of primes π it possesses a Hall π-subgroup (a subgroup whose order and index are coprime, with order the full π-part of |G|). This generalizes Sylow existence from a single prime to arbitrary prime sets.

Related classes

Each of these conditions implies solvable (all inclusions strict):

cyclic ⊂ abelian ⊂ nilpotent ⊂ supersolvable ⊂ solvable

• Every abelian, nilpotent, supersolvable, or p-group is solvable.
• Metabelian means derived length ≤ 2 (an abelian extension of an abelian group); metabelian ⊂ solvable, but is neither contained in nor contains nilpotent.

Examples

• S₃ — derived series S₃ ▷ A₃ ▷ 1 (with A₃ ≅ C₃); derived length 2, metabelian.
• S₄ — derived series S₄ ▷ A₄ ▷ V ▷ 1, where V ≅ C₂ × C₂ is the Klein four-group; derived length 3. Solvable but not supersolvable.
• Every dihedral Dₙ — metacyclic, hence solvable.
• Every group of order less than 60 is solvable — 60 is the smallest order admitting a non-solvable group.

Non-examples

• A₅ — order 60, the smallest non-solvable group and simultaneously the smallest non-abelian simple group. It is its own only nontrivial composition factor, which is non-abelian.
• S₅ — order 120; its composition factors are A₅ and C₂, so the non-abelian A₅ blocks solvability.
• SL(2,𝔽5) — order 120, perfect (SL(2,𝔽5)′ = SL(2,𝔽5)), with composition factor A₅ ≅ PSL(2,𝔽5).
• Any finite group with a non-abelian simple composition factor — e.g. every non-abelian simple group, and the matrix groups GL(2,𝔽q), SL(2,𝔽q) for q ≥ 4.

See also

• Nilpotent group — the stronger central-series condition.
• Supersolvable group — cyclic-factor chief series.
• Metabelian group — derived length ≤ 2.
• Derived subgroup — the derived series that defines solvability.
• Simple group — a non-abelian simple factor is the sole obstruction to solvability.
• Burnside's theorem, Hall subgroup.
• Wikipedia: Solvable group