# Free nilpotent group

From Groupprops

## Contents

## Definition

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.

## Arithmetic functions

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:

Function | Value | Explanation |
---|---|---|

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 |

## Group properties

Property | Meaning | Satisfied? | Explanation |
---|---|---|---|

nilpotent group | Yes | ||

UL-equivalent group | upper central series and lower central series coincide | Yes |

## Particular cases

- 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).

### Hirsch lengths

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) |
---|---|---|

1 | ||

2 | ||

3 | ||

4 |