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The quotient group of the additive abelian group of rational numbers by its subgroup of integers.
It is also the coequalizer of the identity function of the rational numbers and the function on the rational numbers:
For each natural number the cyclic group is a subgroup
In fact is the filtered colimit of a diagram of cyclic groups and inclusions between them. In more detail, whenever , multiplication by induces an inclusion
and this gives a functor to abelian groups from the lattice of natural numbers ordered by divisibility. The colimit of this functor is the abelian group .
For each prime the -Prüfer group , similarly formed as the colimit of groups and inclusions between them, embeds in . Furthermore, it follows from the Chinese remainder theorem? that the induced map
is an isomorphism.
We have a canonical subgroup inclusion into the circle group: If the latter is identified as the canonical subgroup in the group of units of the complex numbers, this inclusion is given by
From this point of view, is the torsion subgroup of U(1), whose elements are precisely the roots of unity.
The group is an injective object in the category Ab of abelian groups.
It is also a cogenerator in the category of abelian groups.
Proof: let be an abelian group. It suffices to check that for every there exists such that . But if is the cyclic subgroup generated by , then it is easy to find a map such that , and then we can extend to a map using injectivity of .
This means every abelian group embeds into an injective abelian group,
and into an algebraic double dual, . The algebraic double dual of is its profinite completion
where is the group of -adic integers (this is connected with the direct sum decomposition of into Prüfer -groups).
Last revised on December 9, 2023 at 12:14:10. See the history of this page for a list of all contributions to it.