Exercise 13, § 10

Let E be a right vector space of dimension $\geqslant 2$ and $H_0$ a hyperplane of E.

(a) Show that every automorphism u belonging to F(E) (Exercise 12) is the product of an automorphism of $\mathbf{SL}(E)$ and possibly a dilatation of the hyperplane $H_0$ (proceed by induction on the codimension of the kernel of $1_E - u$, using Exercises 12(a) and 12(f)).

(b) Show that $\mathbf{SL}(E)$ contains the commutator group of F(E) and is identical with this group except when E is a space of dimension 2 over the field with 2 elements. (To show that $\mathbf{SL}(E)$ contains the commutator group of F(E), use (a) and Exercise 12(f). To see that $\mathbf{SL}(E)$ is contained in the commutator group of F(E), except in the case indicated, show that for every homomorphism of F(E) into a commutative group, the image of a transvection is the identity element, using Exercises 11(b) and 12(c).)

(c) Show that $\mathbf{SL}(E)$ is equal to its commutator group except when dim E = 2 and K has 2 or 3 elements. (If dim E $\geqslant 3$, note that every transvection u can be written as $vvw^{-1}w^{-1}$, where v is a transvection and $w \in \mathbf{SL}(E)$, using Exercise 12(d). If dim E = 2, note first that every automorphism u whose matrix with respect to a basis of E is of the form $\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}$ is a product of transvections, arguing as in (a); then consider the commutator $uvu^{-1}v^{-1}$, where v is the transvection whose matrix with respect to the same basis is $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.)