Additive Group of Integers is Normal 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
From Additive Group of Integers is Subgroup of Rationals, $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.

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 $\struct {\Q, +}$.