Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup/Examples/Non-Zero Integers in Non-Zero Reals under Multiplication
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 Infinite Abelian Group for further confirmation.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $4$