Group of rational numbers with square-free denominators

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Definition

The group of rational numbers with square-free denominators is defined in the following equivalent ways:

  1. It is the subgroup of the group of rational numbers comprising those rational numbers whose denominators are square-free numbers, i.e., no square of a prime should divide the denominator.
  2. It is the join of the following subgroups of the group of rational numbers: for each prime number p, consider the cyclic subgroup generated by the element 1/p.
  3. As an abstract group, it is isomorphic to the quotient group of the restricted external direct product of countably many copies of the group of integers, with the copies indexed by prime numbers, by the identification of the prime times the group generator. Explicitly, if p_1,p_2,\dots are the primes and \mathbb{Z}_1,\mathbb{Z}_2,\dots are the direct factors for the respective p_is, we want to set the p_i \in \mathbb{Z}_i to equal the p_j \in \mathbb{Z}_j for all i,j.

Group properties

Property Satisfied? Explanation
cyclic group No
finitely generated group No
locally cyclic group Yes
abelian group Yes
group whose automorphism group is cyclic Yes

Utility as example