nilpotent group

A nilpotent group is one that is “almost abelian” in a controlled, layered way: its failure to be commutative dissolves if you repeatedly fold in the center. Each time you quotient out by commutators relative to the whole group, the group gets simpler, and for a nilpotent group this process reaches the trivial group in a bounded number of steps.

Formally, the lower central series of G is defined by γ_1 = G and γ_{i+1} = [G, γ_i], the subgroup generated by all commutators of a group element with an element of γ_i. The group is nilpotent if γ_{c+1} = 1 for some c; the least such c is its nilpotency class. Equivalently the upper central series 1 ⊴ Z(G) ⊴ Z_2(G) ⊴ ... (each step the preimage of the center of the quotient) reaches all of G in finitely many steps.

Nilpotent sits strictly between abelian and solvable: every abelian group is nilpotent of class ≤ 1, and every nilpotent group is solvable, but neither converse holds. For finite groups nilpotency has a clean characterization — a finite group is nilpotent if and only if it is the direct product of its Sylow subgroups, equivalently if every Sylow subgroup is normal. This is why finite p-groups are the prototypical nilpotent groups.