Group of integers in group of rational numbers

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This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) group of integers and the group is (up to isomorphism) group of rational numbers (see subgroup structure of group of rational numbers).
The subgroup is a normal subgroup and the quotient group is isomorphic to group of rational numbers modulo integers.
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Definition

This page is about the situation where the whole group is the group of rational numbers (\mathbb{Q},+) (under addition) and the subgroup involved is the group of integers (\mathbb{Z},+).

Note that if we only care about the subgroup up to automorphism, we can pick the cyclic subgroup generated by any nonzero element of the whole group.

The quotient group \mathbb{Q}/\mathbb{Z}, the group of rational numbers modulo integers, is a group in its own right.

Effect of subgroup operators

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer the whole group -- group of rational numbers
centralizer the whole group -- group of rational numbers
normal core the subgroup itself current page group of integers
normal closure the subgroup itself current page group of integers
characteristic core trivial subgroup -- trivial group
characteristic closure whole group -- group of rational numbers

Subgroup properties

Invariance properties

Property Meaning Satisfied? Explanation Comment
normal subgroup invariant under all inner automorphisms Yes on account of being a subgroup of abelian group.
powering-invariant subgroup whenever the group is powered over a prime, so is the subgroup. No \mathbb{Q} is powered over all primes, \mathbb{Z} over none. It is therefore also not a divisibility-closed subgroup or a quotient-powering-invariant subgroup.
characteristic subgroup invariant under all automorphisms No follows from not being powering-invariant (see characteristic subgroup of abelian group implies powering-invariant) on account of this, it is also not a fully invariant subgroup and it does not satisfy any of the other properties stronger than characteristicity.
Note also that the group of rational numbers, on account of being the additive group of a field, is a group whose automorphism group is transitive on non-identity elements, hence a characteristically simple group.

Other resemblance properties

Property Meaning Satisfied? Explanation Comment
isomorph-automorphic subgroup it is automorphic to any other subgroup that it is isomorphic to.