Exercise 9, § 5

Let $V$ be a vector space of dimension 4 over a finite field $K$, of characteristic different from 2, and let $Q$ be a non-degenerate quadratic form on $V$ of index 2 (\emph{Algebra}, Chap. IX, § 4, no. 2). Let $G = O(Q)$ be the orthogonal group of this form; this is a finite group generated by reflections (\emph{loc. cit.}, § 6, no. 4, Prop. 5).

a) Let $E$ be a maximal totally isotropic subspace of $V$, and let $G_E$ be the subgroup of $G$ consisting of the elements $g$ such that $g(x) = x$ for all $x \in E$. Show that $G_E$ is isomorphic to the additive group of $K$, and contains no pseudo-reflection.

b) Show that the algebra of symmetric invariants of $G$ is not a graded polynomial algebra (use the preceding exercise).