# One-relator 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

### Symbol-free definition

A group is said to be a **one-relator group** if it satisfies the following equivalent conditions:

- It has a presentation with only one relation
- It is the quotient of a free group by a point-closure (viz, the normal closure of a single element)