Multiplicative Group of Rationals Subgroup of Reals

Theorem

Let $\left({\Q^*, \times}\right)$ be the Multiplicative Group of Rational Numbers.

Let $\left({\R^*, \times}\right)$ be the Multiplicative Group of Real Numbers.

Then $\left({\Q^*, \times}\right)$ is a normal subgroup of $\left({\R^*, \times}\right)$.

Proof

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

As $\left({\R^*, \times}\right)$ is a group, and $\left({\Q^*, \times}\right)$ is a group, it follows from the definition of subgroup that $\left({\Q^*, \times}\right)$ is a subgroup of $\left({\R^*, \times}\right)$.

As $\left({\R^*, \times}\right)$ is abelian, it follows from All Subgroups of Abelian Group are Normal that $\left({\Q^*, \times}\right)$ is normal in $\left({\R^*, \times}\right)$.

$\blacksquare$

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 36 \ (1)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 4$: Example $4.3$