# Group of rational numbers modulo integers

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

This group, denoted , is defined in the following equivalent ways:

- It is the quotient of the additive group of rational numbers by the subgroup that is the additive group of integers .
- It is the subgroup of the multiplicative group of complex numbers comprising all the roots of unity.
- It is the restricted external direct product of the -quasicyclic groups for all prime numbers .

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

cyclic group | No | |

periodic group (torsion group) | Yes | every rational number has an integer multiple that is an integer. Thus, in the quotient group, it has finite order. |

locally cyclic group | Yes | |

abelian group | Yes | |

epabelian group | Yes | |

divisible abelian group | Yes | |

group whose automorphism group is abelian | Yes | locally cyclic implies aut-abelian |

## Topology

If we start with the usual Euclidean topologies on and and give the group the quotient topology, it is a topological group and in particular a T0 topological group. We can think of it as a dense subgroup inside the circle group, which we can consider to be .