Difference between revisions of "Recursively presented group"

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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]], viz a [[defining ingredient::presentation of a group|presentation]] where the number of generators is finite and the set of relations is recursively enumerable.
+
# 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]], viz a [[defining ingredient::presentation of a group|presentation]] where the number of generators is finite and the set of relations is recursive.
+
# 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 is a [[defining ingredient::finitely generated group]] that is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]].
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# 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::Finite group]] || || || || {{intermediate notions short|recursively presentable group|finite group}}
+
| [[Weaker than::finite group]] || || || || {{intermediate notions short|recursively presentable group|finite group}}
 
|-
 
|-
| [[Weaker than::Finitely presentable group]] || || || || {{intermediate notions short|recursively presentable group|finitely presentable group}}
+
| [[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
 
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
|-
 
|-
| [[Stronger than::Finitely generated group]] || || || || {{intermediate notions short|finitely generated group|recursively presentable group}}
+
| [[Stronger than::countable group]] || || || || {{intermediate notions short|countable group|recursively presentable group}}
|-
 
| [[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 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, i.e., a presentation where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursively enumerable.
  2. It possesses a recursive presentation, i.e., a presentation where the number of generators is finite and the set of relations is recursive.
  3. 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