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 All Subgroups of Abelian Group are Normal it follows that $$\left({\Q, \times}\right) \triangleleft \left({\C, \times}\right)$$.