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

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

$\struct {\Z_{\ne 0}, \times}$ in $\struct {\R_{\ne 0}, \times}$

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:

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