Additive Group of Reals is Normal Subgroup of Complex

Theorem

Let $\struct {\R, +}$ be the additive group of real numbers.

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

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

Proof

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

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

The result follows from Subgroup of Abelian Group is Normal.

$\blacksquare$