Non-Zero Real Numbers under Multiplication form Abelian Group/Proof 2

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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

Because Real Numbers under Multiplication form Monoid and the non-zero numbers are exactly the invertible elements of real multiplication (Inverses for Real Multiplication) then the non zero reals under multiplication form a group by Invertible Elements of Monoid form Subgroup of Cancellable Elements.

The group must also be Abelian because Real Multiplication is Commutative and Subset Product within Commutative Structure is Commutative

$\blacksquare$


Sources