Multiplicative Group of Real Numbers

Theorem

Let $\R^*$ be the set of real numbers without zero, i.e. $\R^* = \R \setminus \left\{{0}\right\}$.

The structure $\left({\R^*, \times}\right)$ is an infinite abelian group.

Proof

Taking the group axioms in turn:

G0: Closure

Real Multiplication is Closed.

G1: Associativity

Real Multiplication is Associative.

G2: Identity

The identity element of real number multiplication is the real number $1$:

$\exists 1 \in \R: \forall a \in \R: a \times 1 = a = 1 \times a$

G3: Inverses

Each element $x$ of the set of non-zero real numbers $\R^*$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication:

$\forall x \in \R^*: \exists \dfrac 1 x \in \R^*: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$

C: Commutativity

Real Multiplication is Commutative.

Infinite

Real Numbers are Uncountably Infinite.

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.5$: Example $82$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.2$
• Ian D. Macdonald: The Theory of Groups (1968): $\S 1$: Example $1.06$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 33, \S 34 \ (1)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.5$