General affine group:GA(1,Q)

Definition
This group, denoted $$GA(1,\mathbb{Q})$$ or $$AGL(1,\mathbb{Q})$$, is defined in the following equivalent ways:


 * 1) It is the member of family::general affine group of degree one over the field of rational numbers $$\mathbb{Q}$$. Explicitly, it is the group of transformations of the form $$x \mapsto ax + b$$, where $$a \in \mathbb{Q}^*, b \in \mathbb{Q}$$, with multiplication defined by composition.
 * 2) It is the defining ingredient::holomorph of the (additive) group of rational numbers.
 * 3) It is the external semidirect product $$\mathbb{Q} \rtimes \mathbb{Q}^*$$ where the latter acts on the former by multiplication.