Non-Zero Real Numbers under Multiplication form Group

Theorem

Let $\R_{\ne 0}$ be the set of real numbers without zero:

$\R_{\ne 0} = \R \setminus \set 0$

The structure $\struct {\R_{\ne 0}, \times}$ forms a group.

Proof

Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

From Non-Zero Real Numbers Closed under Multiplication: Proof 2, $\R_{\ne 0}$ is closed under multiplication.

Note that proof 2 needs to be used specifically here, as proof 1 rests on this result.

$\Box$

Group Axiom $\text G 1$: Associativity

Real Multiplication is Associative.

$\Box$

Group Axiom $\text G 2$: Existence of Identity Element: Real Multiplication Identity is One

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

$\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$

$\Box$

Group Axiom $\text G 3$: Existence of Inverse Element: Inverse for Real Multiplication

Each element $x$ of the set of non-zero real numbers $\R_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication:

$\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$

$\blacksquare$

Sources

• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP: Example $5$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 33$. The definition of a group