Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup/Examples/Non-Zero Integers in Non-Zero Reals under Multiplication

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Examples of Use of Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup

Let $\struct {\Z_{\ne 0}, \times}$ denote the algebraic structure formed by the set of non-zero integers under multiplication.

Let $\struct {\R_{\ne 0}, \times}$ denote the algebraic structure formed by the set of non-zero real numbers under multiplication.


Let $H$ denote the set:

$H := \set {x \times y^{-1}: x, y \in \Z_{\ne 0} }$

Then $H$ is the subgroup of $\struct {\R_{\ne 0}, \times}$ which is the set of non-zero rational numbers under multiplication:

$H = \struct {\Q_{\ne 0}, \times}$


Proof

From Non-Zero Integers Closed under Multiplication:

$\forall a, b \in \Z_{\ne 0}: a \times b \in \Z_{\ne 0}$

From Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup:

$H$ is a subgroup of $\struct {\R_{\ne 0}, \times}$

The elements of $H$ are all the numbers of the form:

$x = \dfrac p q$

where $p, q \in \Z_{\ne 0}$

and thus by definition rational numbers.

Hence the result.

We also note that Non-Zero Rational Numbers under Multiplication form Abelian Group for further confirmation.

$\blacksquare$


Sources