Additive Group of Rationals is Subgroup of Complex

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Theorem

Let $\left({\Q, +}\right)$ be the additive group of rational numbers.

Let $\left({\C, +}\right)$ be the additive group of complex numbers.


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


Proof

From Additive Group of Rationals is Subgroup of Reals, $\left({\Q, +}\right) \lhd \left({\R, +}\right)$.

From Additive Group of Reals is Subgroup of Complex, $\left({\R, +}\right) \lhd \left({\C, +}\right)$.

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

From Complex Numbers under Addition form Abelian Group, $\left({\C, +}\right)$ is abelian.

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

$\blacksquare$