Group of integers in group of rational numbers

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.
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groups

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.