Affine Group of One Dimension as Semidirect Product

Theorem
Let $\operatorname{Af}_1 \left({\R}\right)$ 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 $\phi: \R^\times \to \operatorname{Aut} \left({\R^+}\right)$ be defined as:
 * $\forall b \in \R^\times: \phi \left({b}\right) = \left({a \mapsto a b}\right)$

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

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

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

Then:

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

Then:

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

Also see

 * Affine Group as Semidirect Product