Finitely generated nilpotent group

Definition
A finitely generated nilpotent group is a group satisfying the following equivalent conditions:


 * 1) It is finitely generated and nilpotent.
 * 2) It is finitely presented and nilpotent.
 * 3) It is Noetherian (i.e., every subgroup is finitely generated -- this is also described using the adjective "slender") and nilpotent.
 * 4) It is nilpotent and its abelianization is finitely generated.