Free nilpotent group

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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

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}