# UL-equivalent group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A group is said to be **UL-equivalent**if it is nilpotent and its upper central series and lower central series actually coincide. In other words, if the nilpotence class is , the term of the Upper central series (?) equals the term of the Lower central series (?). (Note that we start counting the lower central series from at the group, and the upper central series at from the trivial subgroup).

### Definition with symbols

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Abelian group | Nilpotency class is 0 or 1 | abelian implies UL-equivalent | UL-equivalent not implies abelian (see also list of examples) | |FULL LIST, MORE INFO |

Special group | Group of prime power order whose center, derived subgroup, and Frattini subgroup coincide (according to some conventions, we also add in the elementary abelian groups) | |FULL LIST, MORE INFO | ||

Extraspecial group | Special group where the center is cyclic | (via special) | (via special) | |FULL LIST, MORE INFO |

Maximal class group | Group of order , nilpotency class , with , prime | maximal class implies UL-equivalent | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Nilpotent group | (by definition) | nilpotent not implies UL-equivalent (see also list of examples) | |FULL LIST, MORE INFO |