# Epinilpotent 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

## Contents

## Definition

A group is termed **epinilpotent** if there is a nonnegative integer such that it satisfies the following equivalent conditions:

- The class c-epicenter of equals the whole group .
- The only quotient group of that is a class c-capable group is the trivial group.

The smallest satisfying any of these equivalent conditions is termed the *epinilpotency class*.

Note that finitely generated and epinilpotent implies cyclic, so among finitely generated groups, the only epinilpotent group are cyclic, hence epabelian, so they have epinilpotency class one.

## Relation with other properties

### Stronger properties

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

epabelian group | epinilpotency class one | |||

cyclic group | via epabelian | |||

locally cyclic group | via epabelian | |||

periodic divisible abelian group | via epabelian |

### Weaker properties

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

nilpotent group |