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:

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

Function	Value	Explanation
nilpotency class	$c$	by construction (note that the class being exactly $c$ is relatively easy to check, and follows also from the existence of any group with class exactly $c$).
Hirsch length	$\sum _{r=1}^{c}\left({\frac {1}{r}}\sum _{d|r}\mu (d)n^{r/d}\right)$	We have to add up the ranks of all the successive factor groups in the lower central series.
minimum size of generating set	$n$	by construction

Group properties

Property	Meaning	Satisfied?	Explanation
nilpotent group		Yes
UL-equivalent group	upper central series and lower central series coincide	Yes

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.

Nilpotency class $c$	Rank of center as a free abelian group (given by formula for dimension of graded component of free Lie algebra)	Hirsch length (equals sum of rank of center and Hirsch length of preceding group)
1	$n$	$n$
2	${\frac {n^{2}-n}{2}}$	${\frac {n(n+1)}{2}}$
3	${\frac {n^{3}-n}{3}}$	${\frac {n(n+1)(2n+1)}{6}}$
4	${\frac {n^{4}-n^{2}}{4}}$	${\frac {n(n+1)(3n^{2}+n+2)}{12}}$