# Upper central series not is strongly central

From Groupprops

## Contents

## Statement

The Upper central series (?) of a Nilpotent group (?) need not be a Strongly central series (?).

## Definitions used

### Upper central series

`Further information: Upper central series`

### Strongly central series

`Further information: Strongly central series`

## Related facts

- Lower central series is strongly central
- Nilpotent not implies UL-equivalent: In a nilpotent group, the upper and lower central series need not be the same.
- Upper central series may be tight with respect to nilpotence class

## Facts used

- Upper central series may be tight with respect to nilpotence class: For any natural number , we can construct a nilpotent group such that if denotes the member of the upper central series, the part of the upper central series upto is also the upper central series of . In particular, has nilpotence class .

## Proof

For , consider a group that fits the situation of fact (1). Then, has class exactly equal to two.

Suppose now that the upper central series of were strongly central. Then, when numbered from downwards, is the member, so by the definition of strongly central, is in the member from downwards, which is the trivial subgroup since for . Thus, is Abelian.

This is a contradiction, so the upper central series of is *not* strongly central.