Additive Group of Integers is Subgroup of Rationals

Theorem
Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\Q, +}$ be the additive group of rational numbers.

Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$.

Proof
Recall that Integers form Integral Domain.

The set $\Q$ of rational numbers is defined as the quotient field of the integers.

The fact that the integers are a subgroup of the rationals follows from the work done in proving the Existence of Quotient Field from an integral domain.

From Rational Numbers under Addition form Abelian Group, $\struct {\Q, +}$ is an abelian group.

From Subgroup of Abelian Group is Normal it follows that $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$.