Multiplicative Group of Rationals is Normal Subgroup of Complex

Theorem
Let $\left({\Q, \times}\right)$ be the multiplicative group of rational numbers.

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

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

Proof
From Multiplicative Group of Rationals Subgroup of Reals, $\left({\Q, \times}\right) \triangleleft \left({\R, \times}\right)$.

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

Thus $\left({\Q, \times}\right) \le \left({\C, \times}\right)$.

As the Multiplicative Group of Complex Numbers is abelian, from Subgroup of Abelian Group is Normal it follows that $\left({\Q, \times}\right) \triangleleft \left({\C, \times}\right)$.