Permutation Group/Examples/Example on Real Numbers

Example of Permutation Group
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 $\struct {S, \oplus}$.

Proof
From Group Example: $\tuple {a c, a d + b}$ on Positive Reals by Reals, $\struct {S, \oplus}$ has been shown to be a group.

From Bijection Example: $a x + b$ on Real Numbers, we have that $f_{a, b}$ is a bijection from $\R$ to $\R$.

Hence by definition $f_{a, b}$ is a permutation on $\R$.

From Group Example: Linear Functions, $\struct {\GG, \circ}$ is a group.

Hence as $\struct {\GG, \circ}$ is a group of permutations on $\R$, it is by definition a permutation group on $\R$.

Let $\phi: \struct {\GG, \circ} \to \struct {S, \oplus}$ be the identity mapping on $S$:


 * $\forall \tuple {a, b} \in S: \map \phi {a, b} = \tuple {a, b}$

From Identity Mapping is Bijection we have that $\phi$ is a bijection.

Consider $f_{a, b}, f_{c, d} \in \GG$.

We have:

This demonstrates that $\phi$ is a (group) homomorphism.

So $\phi$ is a bijection which is a homomorphism.

Hence by definition $\phi$ is an isomorphism.

The result follows.