Additive Group of Reals 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.

$\blacksquare$

Sources

• John F. Humphreys: A Course in Group Theory (1996): $\S 4$: Example $4.3$