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.

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