Recursively presented group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[group]] is said to be '''recursively presented''' if it | 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]], 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]], 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 recursive. | |||
# 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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===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::finite group]] || || || || {{intermediate notions short|recursively presentable group|finite group}} | |||
|- | |||
| [[Weaker than::finitely presented group]] || || || || {{intermediate notions short|recursively presented group|finitely presented group}} | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::countable group]] || || || || {{intermediate notions short|countable group|recursively presentable group}} | |||
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Latest 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:
- 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 |