Exercise 11, § 10

Let $E$ be a right vector space over a field $K$, of dimension $\geq 1$, and $H$ a hyperplane of $E$. Every endomorphism $u$ of $E$ leaving invariant each of the elements of $H$ gives, on passing to the quotients, an endomorphism of the 1-dimensional quotient space $E/H$, an endomorphism which is therefore of the form $\dot{x} \mapsto \dot{x} \mu(\dot{x})$, where $\mu(\dot{x}) \in K$ is such that $\mu(\dot{x}\lambda) = \lambda^{-1} \mu(\dot{x}) \lambda$ for $\lambda \in K^*$. An automorphism of $E$ leaving invariant the elements of $H$ is called a transvection of hyperplane $H$ if the corresponding automorphism of $E/H$ is the identity and a dilatation of hyperplane $H$ otherwise; when $u$ is a dilatation, the set of elements $\mu(\dot{x})$ for $\dot{x} \in E/H$ which is a class of conjugate elements in the multiplicative group $K^*$, is called the class of the dilatation $u$.

(a) Show that for every dilatation there exists one and only one supplementary line of H, invariant under the dilatation.

(b) Let $\phi$ be a linear form on E such that $H = \phi^{-1}(0)$; show that for every transvection $u$ of hyperplane H there exists a unique vector $a \in H$ such that $u(x) = x + a\phi(x)$ for all $x \in E$. Let $\Gamma(E, H)$ be the subgroup of $\mathbf{GL}(E)$ consisting of the automorphisms leaving invariant each element of H; show that the transvections of hyperplane H form a normal commutative subgroup $\Theta(E, H)$ of $\Gamma(E, H)$ isomorphic to the additive group H; the quotient group $\Gamma(E, H)/\Theta(E, H)$ is isomorphic to the multiplicative group $K^*$. For $\Gamma(E, H)$ to be commutative, it is necessary and sufficient that one of the two following conditions be satisfied: ($\alpha$) K is a field with two elements; ($\beta$) K is commutative and $\dim E = 1$.

(c) Suppose that E is finite-dimensional; show that for every transvection $u$ there exists a basis of E such that the matrix of $u$ with respect to this basis has all its diagonal elements equal to 1 and at most one other element $\neq 0$.

(d) Show that the centralizer (I, § 5, no. 3) of the group $\Theta(E, H)$ in the group $\mathbf{GL}(E)$ is the composition $Z(E)\Theta(E, H) = \Theta(E, H)Z(E)$ of $\Theta(E, H)$ and the centre $Z(E)$ of $\mathbf{GL}(E)$ (cf. § 1, Exercise 26). The only automorphisms belonging to this centralizer and leaving invariant at least one element $\neq 0$ of E are the transvections of the group $\Theta(E, H)$. If K has at least 3 elements, the centralizer of $\Gamma(E, H)$ in $\mathbf{GL}(E)$ is equal to $Z(E)$.

(e) Show that the normalizers (I, § 5, no. 3) of $\Theta(E, H)$ and $\Gamma(E, H)$ in $\mathbf{GL}(E)$ are both equal to the subgroup consisting of the automorphisms leaving H invariant.