Group of rational numbers modulo integers

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


 * 1) It is the quotient of the additive defining ingredient::group of rational numbers $$\mathbb{Q}$$ by the subgroup that is the additive defining ingredient::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 $$p$$.

Topology
If we start with the usual Euclidean topologies on $$\mathbb{Q}$$ and $$\mathbb{Z}$$ and give the group the quotient topology, it is a has structure of::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 $$\R/\mathbb{Z}$$.