# Difference between revisions of "Recursively presented group"

From Groupprops

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A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it satisfies the following equivalent conditions: | A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it satisfies the following equivalent conditions: | ||

− | # It possesses a [[defining ingredient::recursive presentation]], | + | # It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group|presentation]] where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursively enumerable. |

− | # It possesses a [[defining ingredient::recursive presentation]], | + | # It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group|presentation]] where the number of generators is finite and the set of relations is recursive. |

− | # It | + | # It is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]]. |

==Relation with other properties== | ==Relation with other properties== | ||

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

|- | |- | ||

− | | [[Weaker than:: | + | | [[Weaker than::finite group]] || || || || {{intermediate notions short|recursively presentable group|finite group}} |

|- | |- | ||

− | | [[Weaker than:: | + | | [[Weaker than::finitely presented group]] || || || || {{intermediate notions short|recursively presented group|finitely presented group}} |

|} | |} | ||

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

|- | |- | ||

− | | [[Stronger than:: | + | | [[Stronger than::countable group]] || || || || {{intermediate notions short|countable group|recursively presentable group}} |

− | |||

− | |||

|} | |} |

## Revision as of 02:01, 5 November 2013

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 number of generators is finite 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 |