Exercise 6, § 9

(a) Let E be an affine space of dimension $\geqslant 2$ over a field K. For an affine mapping $u$ of E into itself to transform every line of E into a parallel line, it is necessary and sufficient that the linear mapping $v$ associated with $u$ be a homothety $t \mapsto \gamma t$ of ratio $\gamma \neq 0$ belonging to the centre of K. If $\gamma = 1$, $u$ is a translation; show that if $\gamma \neq 1$, there exists one and only one point $a \in E$ such that $u(a) = a$. If $a$ is taken as origin of E, $u$ is then identified with a central homothety for the vector space structure thus determined on E; $u$ is called a central homothety of the affine space E of centre $a$ and ratio $\gamma$.

(b) Let $u_1, u_2$ be two affine mappings of E into E each of which is either a translation or a central homothety. Show that $u_1 \circ u_2$ is a translation or a central homothety of E; if $u_1, u_2$ and $u_1 \circ u_2$ are all three central homotheties show that their centres lie on a straight line. What can be said when $u_1$ and $u_2$ are central homotheties and $u_1 \circ u_2$ is a translation?

(c) Show that the set of translations and central homotheties is a normal subgroup H of the affine group of E and that H/T is isomorphic to the multiplicative group of the centre of K: show that H can only be commutative if H = T, in other words if the centre of K has only two elements.