Group of rational numbers modulo integers

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

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

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 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 $\mathbb {R} /\mathbb {Z}$.