# Non-Zero Real Numbers under Multiplication form Abelian Group

## Theorem

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

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

The structure $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.

## Proof 1

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:

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

$\Box$

### $\text C$: Commutativity

Real Multiplication is Commutative.

$\Box$

### Infinite

Real Numbers are Uncountably Infinite.

$\blacksquare$

## Proof 2

We have Real Numbers under Multiplication form Monoid.

From Inverse for Real Multiplication, the non-zero numbers are exactly the invertible elements of real multiplication.

Thus from Invertible Elements of Monoid form Subgroup of Cancellable Elements, the non-zero real numbers under multiplication form a group.

From:

it follows that this group is also Abelian.

$\blacksquare$

## Proof 3

From Non-Zero Real Numbers under Multiplication form Group, $\struct {\R_{\ne 0}, \times}$ forms a group.

$\Box$

From Real Multiplication is Commutative it follows that $\struct {\R_{\ne 0}, \times}$ is abelian.

$\Box$

From Real Numbers are Uncountably Infinite it follows that $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.2$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.06$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(1)$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(ii)}$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.5$

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- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups