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
QUICK PHRASES: countable, countably generated
A group is said to be countable or denumerable or enumerable if it satisfies the following equivalent conditions:
- It is countable as a set, i.e., its order is a cardinal that is at most the cardinality of the natural numbers.
- 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
|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|