Additive Group of Integers is Normal Subgroup of Complex

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

Let $\left({\C, +}\right)$ be the Additive Group of Complex Numbers.

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

Proof
From Additive Group of Integers Subgroup of Reals, $\left({\Z, +}\right) \triangleleft \left({\R, +}\right)$.

From Additive Group of Reals Subgroup of Complex, $\left({\R, +}\right) \triangleleft \left({\C, +}\right)$.

Thus $\left({\Z, +}\right) \le \left({\C, +}\right)$.

As the Additive Group of Complex Numbers is abelian, from All Subgroups of Abelian Group are Normal it follows that $\left({\Z, +}\right) \triangleleft \left({\C, +}\right)$.