Additive Group of Reals is Normal Subgroup of Complex

Theorem
Let $$\left({\R, +}\right)$$ be the Additive Group of Real Numbers.

Let $$\left({\C, +}\right)$$ be the Additive Group of Complex Numbers.

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

Proof
Let $$x, y \in \C$$ such that $$x = x_1 + 0 i, y = y_1 + 0 i$$.

As $$x$$ and $$y$$ are wholly real, we have that $$x, y \in \R$$.

Then $$x + y = \left({x_1 + y_1}\right) + \left({0 + 0}\right)i$$ which is also wholly real.

Also, the inverse of $$x$$ is $$-x = -x_1 + 0 i$$ which is also wholly real.

Thus by the Two-Step Subgroup Test, $$\left({\R, +}\right)$$ is a subgroup of $$\left({\C, +}\right)$$.

Then we note that $\left({\C, +}\right)$ is abelian.

The result follows from All Subgroups of Abelian Group are Normal.