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
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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