GAPlus(1,R)

Definition
This group, denoted $$GA^+(1,\R)$$, is defined as the group (under composition) of functions $$\R \to \R$$ of the form:

$$x \mapsto ax + b, a,b \in \R, a > 0$$

Equivalently, it can be defined as the external semidirect product $$\R \rtimes (\R^*)^+$$ where the latter is the multiplicative group of positive reals and it act on the former by multiplication.

Note that, via the logarithm map, the acting group is isomorphic to $$\R$$. Thus, the group can be defined as $$\R \rtimes \R$$ where $$t \in \R$$ (as an element of the acting group) acts on $$v \in \R$$ (as an element of the base group) by $$t \cdot v = e^tv$$.

The group is a subgroup of index two in general affine group:GA(1,R).