Exercise 13, § 3

Let $G$ be a finite group whose order $n$ is a power $p^f$ of a prime number $p$ and $K$ a field of characteristic $p$.

(a) Let $E$ be a finite set on which $G$ operates (\emph{Algebra}, Chapter I, § 7, no. 2); if $E'$ is the set of elements of $E$ which are invariant for all $g \in G$, show that $\mathrm{Card}(E) \equiv \mathrm{Card}(E')$ (mod. $p$). Deduce that, if $G$ is not reduced to its identity element $e$, the centre $Z$ of $G$ is not reduced to $e$ (make $G$ operate on itself by inner automorphisms). Conclude that $G$ is solvable (\emph{Algebra}, Chapter I, § 6, Exercise 14).

(b) Let $V$ be a vector space over $K$ and $\rho$ a homomorphism of $G$ to $\mathbf{GL}(V)$. Let $V'$ be the set of $x \in V$ which are invariant under the linear mappings $\rho(g)$, where $g \in G$. Show that, if $V'$ is reduced to 0, $V$ is necessarily reduced to 0. (If $x \in V$ and $x \neq 0$, apply (a) to the additive subgroup $E$ of $V$ generated by the elements $\rho(g)x$, where $g \in G$.)

(c) Let $A = K^{(G)}$ be the algebra of the group $G$ over $K$ (\emph{Algebra}, Chapter 111, § 1) and let $I$ be the vector subspace of $A$ (over $K$) generated by the elements $1 - g$ where $g \in G$. Show that $I$ is the Jacobson radical of $A$ and that $A/I$ is isomorphic to $K$ (use (b) and the definition of Jacobson radical); deduce that $I$ is nilpotent.

(d) Let $V$ be a finitely generated $A$-module and $V'$ the set of elements $x$ of $V$ such that $g.x = x$ for all $g \in G$. Show that the following inequalities hold

() $\dim_K(V) \leq n \cdot \dim_K(V/IV)$; (**) $\dim_K(V) \leq n \cdot \dim_K(V')$.

(To establish (), use Corollary 2 to Proposition 4; deduce (**) by considering the dual of the vector space $V$ over $K$.) For the two sides of the inequality () (resp. (*)) to be equal, it is necessary and sufficient that $V$ be a free $A$-module (cf. Corollary 1 to Proposition 5).