Additive Group of Integers is Subgroup of Rationals

Theorem
Let $$\left({\mathbb{Z}, +}\right)$$ be the Additive Group of Integers.

Let $$\left({\mathbb{Q}, +}\right)$$ be the Additive Group of Rational Numbers.

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

Proof
The integers form an integral domain.

The 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.

The normality of $$\left({\mathbb{Z}, +}\right)$$ in $$\left({\mathbb{Q}, +}\right)$$ follows from the fact that $\left({\mathbb{Q}, +}\right)$ is an abelian group and All Subgroups of Abelian Group are Normal.