Countable 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

QUICK PHRASES: countable, countably generated

A group is said to be countable or denumerable or enumerable if it satisfies the following equivalent conditions:

  1. It is countable as a set, i.e., its order is a cardinal that is at most the cardinality of the natural numbers.
  2. It has a countable generating set.

Sometimes, the term countable is used to refer only to infinite countable groups, i.e., finite groups are specifically excluded. Whether this is the valid interpretation depends on the context.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite group Finitely generated group, Recursively presented group|FULL LIST, MORE INFO
Finitely generated group has a finite generating set |FULL LIST, MORE INFO
Finitely presentable group has a finite presentation Finitely generated group, Recursively presented group|FULL LIST, MORE INFO
Recursively presentable group has a recursive presentation |FULL LIST, MORE INFO