Recursively presented group
From Groupprops
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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Contents
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 number of generators is finite 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 |
