Nilpotent groups

A finite group is nilpotent when it is the direct product of its Sylow subgroups; equivalently, its lower central series reaches the trivial subgroup. The number of steps is its nilpotency class. Every p-group is nilpotent, and every nilpotent group is supersolvable, hence solvable.

Definition

The lower central series of G is

γ₁ = G,   γᵢ₊₁ = [γᵢ, G].

G is nilpotent if γ_{c+1} = {e} for some c; the least such c is the nilpotency class. The upper central series {e} = Z₀ ◁ Z₁ ◁ ⋯, where Zᵢ₊₁/Zᵢ = Z(G/Zᵢ), reaches G in the same number c of steps.

Class 0 — the trivial group.
Class 1 — exactly the nontrivial abelian groups (γ₂ = [G, G] = {e}).
Class ≥ 2 — non-abelian, e.g. D₄ and Q₈ (class 2).

Equivalent characterizations

For a finite group G, the following are equivalent:

the lower central series reaches {e} (γ_{c+1} = {e});
the upper central series reaches G (Z_c = G);
G is the direct product of its Sylow subgroups;
every Sylow subgroup of G is normal;
every maximal subgroup of G is normal — equivalently, has prime index and contains the commutator subgroup [G, G];
the normalizer condition: every proper subgroup H < G is properly contained in its normalizer, H < N_G(H).

The Sylow decomposition is canonical: writing |G| = ∏ p^aₚ,

G  ≅  ∏_p G_p,     G_p = the (unique, normal) Sylow p-subgroup.

Properties

The semi-dihedral group SD₂ⁿ has maximal nilpotency class n − 1 — the largest possible for a 2-group of order 2ⁿ.
The modular maximal-cyclic group is the only nilpotency-class-2 member among the 2-groups of order 2ⁿ with a cyclic index-2 subgroup: M₂ⁿ has class 2, while the dihedral, generalized quaternion, and semi-dihedral groups of that order have maximal class n − 1.
Every nilpotent group is solvable.
Every finite p-group is nilpotent.
Every finite nilpotent group is the direct product of its Sylow subgroups.
A finite group is nilpotent exactly when it is the direct product of its Sylow subgroups: G ≅ ∏_p G_p, one factor per prime.
A finite nilpotent group's subgroup lattice is the direct product of its Sylow subgroups' lattices — abelian groups included.
Nontrivial center. A nontrivial nilpotent group has nontrivial center Z(G) ≠ {e} — in particular every nontrivial finite p-group does. (The upper central series could not climb from {e} to G otherwise.)
Closure. Nilpotent groups are closed under subgroups, quotients, and finite direct products; the class of the product is the max of the factors' classes.
Position among solvable groups. Finite nilpotent ⇒ supersolvable ⇒ solvable. Both converses fail: S₃ is supersolvable but not nilpotent, and A₄ is solvable but not supersolvable.
Fitting subgroup. Every finite group G has a unique largest nilpotent normal subgroup, the Fitting subgroup Fit(G) — the product of all nilpotent normal subgroups. G is nilpotent ⇔ Fit(G) = G.

Examples

All abelian groups — class 1 (Cₙ, C₂ × C₃, …).
All p-groups — e.g. the order-8 non-abelian pair D₄ and Q₈, both class 2. Every group of order 8 is nilpotent.
C₂ × C₃ ≅ C₆ — a direct product of Sylow subgroups of coprime order; class 1.
Any finite nilpotent group splits as the direct product of its Sylow subgroups, so its structure reduces to that of its p-parts. All groups of order 4 are nilpotent.

Non-examples