Permutation Group/Examples

Examples of Permutation Groups

Example on $\R$

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 $\struct {S, \oplus}$.