Supersolvable groups

A finite group is supersolvable when it has a chain of subgroups, each normal in the whole group, with cyclic factors. Equivalently, a finite group whose chief factors all have prime order. Supersolvability sits strictly between nilpotent and solvable.

Definition

A finite group G is supersolvable if it has a normal series

1 = G₀ ◁ G₁ ◁ ⋯ ◁ Gₙ = G

in which

every Gᵢ is normal in G — not merely in Gᵢ₊₁, and
every factor Gᵢ₊₁ / Gᵢ is cyclic.

Equivalently: a finite group with a chief series whose every chief factor has prime order. (A cyclic chief factor of a finite group is automatically some Cₚ.)

A finite group is solvable exactly when it has such a series with cyclic factors that is merely subnormal (each Gᵢ normal only in Gᵢ₊₁); supersolvability strengthens this by demanding each Gᵢ be normal in all of G. Here the cyclic factors are essential: every solvable group already has a normal-in-G series with merely abelian factors — its chief series — so relaxing cyclic to abelian would collapse the condition all the way back to solvable. (A₄ below is the smallest witness. For solvability itself, cyclic and abelian factors are interchangeable.)

In the hierarchy

The following inclusions of classes of finite groups all hold and are strict:

nilpotent  ⊊  supersolvable  ⊊  monomial  ⊊  solvable

together with metacyclic ⊊ supersolvable. Witnesses of the gaps — each lies in the larger class but not the smaller:

S₃ (order 6) — supersolvable, not nilpotent.
C₂ × C₂ × C₂ (order 8) — supersolvable, not metacyclic (its only cyclic normal subgroups have order ≤ 2, with non-cyclic quotient).
A₄ (order 12) — monomial, not supersolvable.
SL(2,𝔽₃) (order 24) — solvable, not monomial (the smallest solvable non-monomial group).

Properties

Subgroup-, quotient-, and product-closed. Every subgroup, every quotient, and every finite direct product of supersolvable groups is supersolvable.
Huppert's maximal-subgroup criterion. A finite group is supersolvable iff every maximal subgroup has prime index.
Sylow tower. Order the prime divisors p₁ > p₂ > ⋯ > p_k. For each j, the Sylow subgroups for p₁, …, p_j together form a normal subgroup of G (the normal Hall {p₁,…,p_j}-subgroup) — so G is assembled by adjoining Sylow subgroups largest-prime first. In particular the Sylow subgroup of the largest prime is normal.
Nilpotent derived subgroup (Huppert). The commutator subgroup [G, G] of a supersolvable group is nilpotent.
Converse of Lagrange. G has a subgroup of order d for every divisor d of |G|.

Examples

Every nilpotent group — hence every p-group and every abelian group. E.g. Q₈, C₄ × C₂.
Every dihedral group Dₙ: the rotation subgroup ⟨r⟩ ≅ Cₙ is normal of index 2, so 1 ◁ ⟨r⟩ ◁ Dₙ refines to a witnessing series.
Every dicyclic group, including the generalized quaternion groups Q₂ⁿ.
Every metacyclic group — a cyclic normal subgroup with cyclic quotient.
Every group of order pq (p, q prime): the Sylow subgroup of the larger prime is normal, so 1 ◁ Cₚ ◁ G has factors Cₚ and C_q. E.g. S₃ (order 6) and the non-abelian C₇ ⋊ C₃ (order 21).

Non-examples

A₄ (order 12) — the smallest solvable group that is not supersolvable. Its chief series 1 ◁ V₄ ◁ A₄ has the non-cyclic chief factor V₄ = C₂ × C₂ of order 4. Equivalently (Huppert), its Sylow 3-subgroup C₃ is a maximal subgroup of index 4 — not prime.
S₄ (order 24) — solvable but not supersolvable: its point-stabilizer maximal subgroup S₃ has index 4, which is not prime.
A₅ (order 60) — simple and non-abelian, so not even solvable, let alone supersolvable.