Group of rational numbers modulo integers
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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 .
|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|
|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 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 .