# Upper central series is fastest ascending central series

## Contents

## Statement

Suppose is a Nilpotent group (?) with a Central series (?) (written in ascending order as):

Denote by the member of the Upper central series (?) of , i.e.:

Then, we have:

In particular, if is the Nilpotence class (?) of , we have:

## Definitions used

### Nilpotent group

### Central series

### Lower central series

### Nilpotence class

## Related facts

## Facts used

## Proof

**Given**: is a Nilpotent group (?) with a Central series (?) (written in ascending order as):

Denote by the member of the Upper central series (?) of , i.e.:

**To prove**:

In particular, if is the Nilpotence class (?) of , we have:

**Proof**: We prove this by induction on .

**Base case for induction**: For , so we're okay.

**Induction step**: Suppose . We want to show that .

Since , and , we see that under the projection , the image of elements of are in the center of . Thus, is in the center of , so so as required.

Since the nilpotence class is the smallest integer for which , must force .