Additive Group of Rationals is Normal Subgroup of Complex

Theorem

Let $\struct {\Q, +}$ be the additive group of rational numbers.

Let $\struct {\C, +}$ be the additive group of complex numbers.

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

Proof

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

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

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

From Complex Numbers under Addition form Infinite Abelian Group, $\struct {\C, +}$ is abelian.

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

$\blacksquare$