Exercise 13, § 10

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

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

b) Now prove that $\mathbf{SL}(E)$ contains the group of commutators of F(E) and is identical with this group unless E is a space of dimension 2 over the field with 2 elements. (To show that $\mathbf{SL}(E)$ contains the group of commutators of $F(E)$, use a) and Exercise 12 $f$). To see that $\mathbf{SL}(E)$ is contained in the group of commutators of $F(E)$ except in the indicated exceptional case, 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) (II, p. 206) and 12 c).

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