Non-Zero Real Numbers under Multiplication form Group
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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