Additive Group of Integers is Subgroup of Rationals

Theorem
Let $\left({\Z, +}\right)$ be the additive group of integers.

Let $\left({\Q, +}\right)$ be the additive group of rational numbers.

Then $\left({\Z, +}\right)$ is a normal subgroup of $\left({\Q, +}\right)$.

Proof
The integers form an 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, $\left({\Q, +}\right)$ is an abelian group.

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