A Classification of Free and Free-Like Nilpotent Groups

Suppose $G$ is a $\mathcal{T}$-group (finitely generated torsion-free nilpotent) with centralizers outside of the derived subgroup being abelian of rank equal to $\text{rank}(Z_1)+1$. This includes the class of free nilpotent groups ${\mathcal{N}}_{r,c}$ of a given rank $r$ and class $c$. It is shown that the upper and lower central series coincide in such groups and from this that they are metabelian. We then prove that all such groups arise as semidirect products of free abelian groups with respect to representation $[G,G]\to \text{UT}(n,\mathbb{Z})$ by automorphisms constructed from integer powers of elements in defining relations we call integral weights of $G$.