Difference between revisions of "Recursively presented group"

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==Definition==
 
==Definition==
  
A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it possesses a [[defining ingredient::recursive presentation]], viz a [[defining ingredient::presentation of a group|presentation]] where:
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A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it satisfies the following equivalent conditions:
  
# The number of generators is finite
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# It possesses a [[defining ingredient::recursive presentation]], viz a [[defining ingredient::presentation of a group|presentation]] where the number of generators is finite and the set of relations is recursively enumerable.
# The set of relations is recursively enumerable
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# It possesses a [[defining ingredient::recursive presentation]], viz a [[defining ingredient::presentation of a group|presentation]] where the number of generators is finite and the set of relations is recursive.
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# It is a [[defining ingredient::finitely generated group]] that is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]].
  
 
==Relation with other properties==
 
==Relation with other properties==

Revision as of 20:07, 26 May 2010

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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

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

  1. It possesses a recursive presentation, viz a presentation where the number of generators is finite and the set of relations is recursively enumerable.
  2. It possesses a recursive presentation, viz a presentation where the number of generators is finite and the set of relations is recursive.
  3. It is a finitely generated group that 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 presentable group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finitely generated group |FULL LIST, MORE INFO
Countable group |FULL LIST, MORE INFO