# Recursively presented 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 said to be **recursively presentable** or **recursively presented** if it satisfies the following equivalent conditions:

- It possesses a recursive presentation, i.e., a presentation where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursively enumerable.
- It possesses a recursive presentation, i.e., a presentation where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursive.
- It is isomorphic to a subgroup of a finitely presented group.

## Relation with other properties

### Stronger properties

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

finite group | |FULL LIST, MORE INFO | |||

finitely presented group | |FULL LIST, MORE INFO |

### Weaker properties

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

countable group | |FULL LIST, MORE INFO |