Free nilpotent group

Definition
Suppose $$c$$ is a positive integer. The free nilpotent group of class $$c$$ on a set $$S$$ can be defined in the following equivalent ways:


 * 1) It is the group $$F/\gamma_{c+1}(F)$$ where $$F$$ is the free group on $$S$$ and $$\gamma_{c+1}(F)$$ is the $$(c+1)^{th}$$ member of the lower central series of $$F$$.
 * 2) It is the reduced free group corresponding to the subvariety of groups of nilpotency class $$c$$ in the variety of groups.

Arithmetic functions
The quotient groups $$\gamma_r(F)/\gamma_{r+1}(F)$$ are free abelian groups of rank given by the formula for dimension of graded component of free Lie algebra. Explicitly, if $$S$$ has size $$n$$, this is:

$$\frac{1}{r} \sum_{d|r} \mu(d)n^{r/d}$$

Particular cases

 * If $$c = 1$$, we get a free abelian group. If $$S$$ is finite and has size $$n$$, this is the group $$\mathbb{Z}^n$$.
 * If $$c = 2$$ and $$S$$ has size two, we get unitriangular matrix group:UT(3,Z).

Hirsch lengths
We work with the group with $$n$$ generators.