# Nilpotent not implies UL-equivalent

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., nilpotent group) neednotsatisfy the second group property (i.e., UL-equivalent group)

View a complete list of group property non-implications | View a complete list of group property implications

Get more facts about nilpotent group|Get more facts about UL-equivalent group

## Contents

## Statement

A nilpotent group need not be UL-equivalent: its Upper central series (?) and Lower central series (?) need not coincide.

## Related facts

### Conditions strong enough to imply UL-equivalent

See UL-equivalent group#Stronger properties.

### Lower central series versus upper central series

Here are some contrasts between facts that are true for lower central series and upper central series. Each of these pairs of facts gives further examples of nilpotent groups that are not UL-equivalent.

Nature of fact | Fact for lower central series | Fact for upper central series |
---|---|---|

Is the series a strongly central series? | Lower central series is strongly central | upper central series not is strongly central (i.e., the upper central series need not always be a strongly central series). |

What is the nilpotency class of the members of the series? | Second half of lower central series of nilpotent group comprises abelian groups, Penultimate term of lower central series is abelian in nilpotent group of class at least three | Upper central series may be tight with respect to nilpotency class |

Are the members verbal subgroups and/or fully invariant subgroups in the whole group? | Lower central series members are verbal (and since verbal implies fully invariant, they are also fully invariant) | Upper central series members need not be fully invariant (even for a nilpotent group) |

## Examples

Below are links to lists of examples:

- All examples on this wiki
- All examples of nilpotency class two
- All examples of nilpotency class three
- All examples of nilpotency class four

## Proof

### A direct product of groups of different nilpotence classes

Suppose are groups of nilpotency classes respectively, with . Then, the direct product is a nilpotent group.

The lower central series and upper central series are both coordinate-wise. Hence, if denotes the term of the lower central series with , then:

Thus, we have:

On the other hand, we have:

In particular, if , then has trivial -projection, but does not.

### Particular examples

The smallest concrete examples of the direct product construction are given below:

Direct product | Order | First factor | Nilpotency class | Second factor | Nilpotency class |
---|---|---|---|---|---|

Direct product of D8 and Z2 | 16 | Cyclic group:Z2 | 1 | Dihedral group:D8 | 2 |

Direct product of Q8 and Z2 | 16 | Cyclic group:Z2 | 1 | Quaternion group | 2 |

Direct product of D8 and Z3 | 24 | Cyclic group:Z3 | 1 | Dihedral group:D8 | 2 |

Direct product of Q8 and Z3 | 24 | Cyclic group:Z3 | 1 | Dihedral group:D8 | 2 |

Some examples that are *not* part of this general construction are nontrivial semidirect product of Z4 and Z4, central product of D8 and Z4, and holomorph of Z8.