Permutation Group/Examples/Example on Real Numbers

Examples of Permutation Groups
Let $\R$ denote the set of real numbers.

Let $\R_{\ge 0}$ denote the set of positive real numbers.

Let $S = \R_{\ge 0} \times \R$ denote the Cartesian product of $\R_{\ge 0}$ and $\R$.

Let $\tuple {a, b} \in S$.

Let $f_{a, b}: R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map {f_{a, b} } x := a x + b$

Let $\GG$ be the set defined as:
 * $\GG = \set {f_{a, b}: \tuple {a, b} \in S}$

Let $\struct {S, \oplus}$ be the group where $\oplus$ is defined as:
 * $\forall \tuple {a, b}, \tuple {c, d} \in S: \tuple {a, b} \oplus \tuple {c, d} := \tuple {a c, a d + b}$

Then $\struct {\GG, \circ}$ is a permutation group on $\R$ which is isomorphic to $\struct {S, \oplus}$.