Affine Group of One Dimension as Semidirect Product

Theorem
Let $\operatorname{Af}_1(\R)$ be the $1$-dimensional affine group on $\R$.

Let $\R^+$ be the additive group of real numbers.

Let $\R^\times$ be the multiplicative group of real numbers.

Let $\phi : \R^\times \to \operatorname{Aut}(\R^+)$ be defined as $\phi(b) = (a \mapsto ab)$ for all $b \in \R^\times$.

Let $\R^+ \rtimes_\phi \R^\times$ be the corresponding semidirect product.

Then:
 * $\operatorname{Af}_1(\R) \cong \R^+ \rtimes_\phi \R^\times$

Proof
Let $f_{ab}, f_{cd} \in \operatorname{Af}_1(\R)$.

Then:

Let $(b,a), (d,c) \in \R^+ \rtimes_\phi \R^\times$.

Then:

So the bijective mapping defined by $f_{ab} \mapsto (b,a)$ is a isomorphism.

Also see

 * Affine Group as Semidirect Product