# Countable 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

View a complete list of group propertiesVIEW 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:

- 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

### 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 |