Multiplicative Group of Rationals is Normal Subgroup of Complex

Theorem

Let $\struct {\Q, \times}$ be the multiplicative group of rational numbers.

Let $\struct {\C, \times}$ be the multiplicative group of complex numbers.

Then $\struct {\Q, \times}$ is a normal subgroup of $\struct {\C, \times}$.

Proof

From Multiplicative Group of Rationals is Normal Subgroup of Reals, $\struct {\Q, \times} \lhd \struct {\R, \times}$.

From Multiplicative Group of Reals is Normal Subgroup of Complex, $\struct {\R, \times} \lhd \struct {\C, \times}$.

Thus $\struct {\Q, \times} \le \struct {\C, \times}$.

From Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group, $\struct {\C, \times}$ is abelian.

From Subgroup of Abelian Group is Normal it follows that $\struct {\Q, \times} \lhd \struct {\C, \times}$.

$\blacksquare$