Additive Group of Integers is Normal Subgroup of Reals

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Theorem

Let $\struct {\Z, +}$ be the additive group of integers.

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


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


Proof

From Additive Group of Integers is Subgroup of Reals, $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.


As the additive group of real numbers is abelian, from Subgroup of Abelian Group is Normal it follows that $\struct {\Z, +} \lhd \struct {\R, +}$.

$\blacksquare$


Sources