Countable group

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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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