Additive Group of Reals is Normal Subgroup of Complex
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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.
$\blacksquare$