# Hypocentral 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

This is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

## Contents

## Definition

A group is said to be **hypocentral** if its lower central series terminates at the identity, or equivalently, if its hypocenter is the trivial group.

## Relation with other properties

### Stronger properties

- Nilpotent group: Here, the lower central series terminates at the identity in finitely many steps, the number of steps being the nilpotence class.
- Residually nilpotent group: Here, the intersection of the finite terms of the lower central series is the trivial group.