# Upper central series

*This article defines a quotient-iterated series with respect to the following subgroup-defining function:* center

## Contents

## Definition

The upper central series of a group is an ascending chain of subgroups indexed by ordinals (including zero), where the member is denoted as . It is defined as follows:

Case for ordinal | Verbal definition of member of upper central series | Definition using the notation |
---|---|---|

trivial subgroup | ||

is a successor ordinal. Note that this includes all positive integer values of . | its image modulo the previous member is the center of the quotient group by the previous member. | is the inverse image of the center of with respect to the natural projection map . In other words, . |

is a limit ordinal. | union of all the previous members | . Equivalently, |

The zeroth member is the trivial group, the first member is the center, and the second member is the second center. In other words:

## Proof methods

The following articles give a sense of the important methods used to prove facts about the upper central series:

- Upward induction for upper central series
- Downward induction for upper central series
- Inductive proof methods for the ascending series corresponding to a subgroup-defining function

## Subgroup properties

### Related subgroup-defining functions

Subgroup-defining function | Role in upper central series |
---|---|

center | first member, i.e., |

second center | second member, i.e., |

hypercenter | the final stable member, i.e., the such that ( may be transfinite; however, it must exist). |

### Related group properties

property | meaning in terms of upper central series |
---|---|

Centerless group | upper central series stays stuck at trivial subgroup; never gets off the ground |

Nilpotent group | upper central series reaches whole group in finitely many steps |

Hypercentral group | transfinite upper central series reaches whole group |

### Relation with lower central series

For a nilpotent group, the lower central series and upper central series are closely related. They both have the same length, and there is a containment relation between them, which follows from the combination of the facts that upper central series is fastest ascending central series and lower central series is fastest descending central series. However, they need not coincide. Nilpotent groups where they do coincide are termed UL-equivalent groups, and nilpotent not implies UL-equivalent.

Here is a table with some distinctions/contrasts between the two central series:

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) |