Free nilpotent group
Suppose is a positive integer. The free nilpotent group of class on a set can be defined in the following equivalent ways:
- It is the group where is the free group on and is the member of the lower central series of .
- It is the reduced free group corresponding to the subvariety of groups of nilpotency class in the variety of groups.
The quotient groups are free abelian groups of rank given by the formula for dimension of graded component of free Lie algebra. Explicitly, if has size , this is:
|nilpotency class||by construction (note that the class being exactly is relatively easy to check, and follows also from the existence of any group with class exactly ).|
|Hirsch length||We have to add up the ranks of all the successive factor groups in the lower central series.|
|minimum size of generating set||by construction|
|UL-equivalent group||upper central series and lower central series coincide||Yes|
- If , we get a free abelian group. If is finite and has size , this is the group .
- If and has size two, we get unitriangular matrix group:UT(3,Z).
We work with the group with generators.
|Nilpotency class||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)|