Non-Zero Real Numbers under Multiplication form Abelian 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 $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.


Proof 1

Taking the group axioms in turn:


$\text G 0$: 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.

$\Box$


$\text G 1$: Associativity

Real Multiplication is Associative.

$\Box$


$\text G 2$: Identity

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

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


$\Box$


$\text G 3$: Inverses

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 Inverses 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:

Real Multiplication is Commutative
Subset Product within Commutative Structure is Commutative

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