Additive Group of Rationals is Normal Subgroup of Reals
Jump to navigation
Jump to search
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 $4.3$