Group of rational numbers with square-free denominators

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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

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 $p_i$s, 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

• Cyclic automorphism group not implies cyclic
• Comes up as the center of the Zalesskii group used by Zalesskii to demonstrate that there exist infinite nilpotent groups in which every automorphism is inner.