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 group
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This group, denoted \mathbb{Q}/\mathbb{Z}, is defined in the following equivalent ways:

  1. It is the quotient of the additive group of rational numbers \mathbb{Q} by the subgroup that is the additive group of integers \mathbb{Z}.
  2. It is the subgroup of the multiplicative group of complex numbers comprising all the roots of unity.
  3. It is the restricted external direct product of the p-quasicyclic groups for all prime numbers p.

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


If we start with the usual Euclidean topologies on \mathbb{Q} and \mathbb{Z} 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 \R/\mathbb{Z}.