# Group with a finite complete rewriting system

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 finitely generated group is termed a **group with a finite complete rewriting system** if it has a complete rewriting system (i.e., a rewriting system that is both finitely terminating and confluent) with respect to a finite generating set such that the rewriting system is finite in size, i.e., it makes use of only finitely many rewriting rules.

Note that it is possible for a group to have a finite complete rewriting system with respect to one finite generating set but not have any finite complete rewriting system with respect to a different finite generating set. We use the term **group with a finite complete rewriting system** if there exists at least one finite generating set for which a finite complete rewriting system exists.

## Relation with other properties

### Stronger properties

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

finitely generated free group | |FULL LIST, MORE INFO | |||

finitely generated abelian group | |FULL LIST, MORE INFO | |||

finitely generated nilpotent group | |FULL LIST, MORE INFO | |||

finite group | |FULL LIST, MORE INFO |

### Weaker properties

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

group with solvable word problem | ||||

finitely generated group |