Recursively presented group

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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 set of generators is countably infinite (with an explicit enumeration) 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