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
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.
$\blacksquare$