Additive Group of Rationals is Normal Subgroup of Reals

Theorem

Let $\struct {\Q, +}$ be the additive group of rational numbers.

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

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

Proof

From the definition of real numbers, $\Q$ is a subset of $\R$.

As $\struct {\R, +}$ is a group, and $\struct {\Q, +}$ is a group, it follows from the definition of subgroup that $\struct {\Q, +}$ is a subgroup of $\struct {\R, +}$.

As $\struct {\R, +}$ is abelian, it follows from Subgroup of Abelian Group is Normal that $\struct {\Q, +}$ is normal in $\struct {\R, +}$.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Example $4.3$