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