Exercise 14, § 10

Let E be a vector space of dimension $\geqslant 2$ over a field K.

(a) Show that the projective group $\mathbf{PGL}(E)$ is canonically isomorphic to the quotient of the linear group $\mathbf{GL}(E)$ by its centre (isomorphic to the multiplicative group of the centre of K, cf. § 1, Exercise 26).

(b) Let $\mathbf{PSL}(E)$ denote the canonical image of the group $\mathbf{SL}(E)$ in $\mathbf{PGL}(E)$; it is a normal subgroup of $\mathbf{PGL}(E)$, containing the commutator group of $\mathbf{PGL}(E)$ when $E$ is finite-dimensional. Show that $\mathbf{PSL}(E)$ is a doubly transitive (cf. I, § 5, Exercise 14) group of permutations of $\mathbf{P}(E)$.

(c) Let $a \neq 0$ be an element of $E$ and let $e = \pi(a) \in \mathbf{P}(E)$ (in the notation of § 9, no. 6). Show that the transvections of the form $x \mapsto x + a \phi(x)$ (where $\phi \in E^*$ is such that $\phi(a) = 0$) constitute a commutative subgroup of $\mathbf{SL}(E)$; if $\Gamma_e$ is the image of this subgroup in $\mathbf{PSL}(E)$, $\Gamma_e$ is a normal subgroup of the subgroup $\Phi_e$ of $\mathbf{PSL}(E)$ leaving $e$ invariant. Show also that the union of the subgroups conjugate to $\Gamma_e$ in $\mathbf{PSL}(E)$ generates $\mathbf{PSL}(E)$.

(d) Let $\Sigma$ be a primitive group (I, § 5, Exercise 13) of permutations of a set $F$ which is equal to its commutator group. Suppose that there exists an element $c \in F$ such that the subgroup $\Phi_c$ of $\Sigma$ leaving $c$ invariant contains a commutative normal subgroup $\Gamma_c$ such that the union of the subgroups conjugate to $\Gamma_c$ in $\Sigma$ generates $\Sigma$. Show that under these conditions $\Sigma$ is simple. (Let $\Delta$ be a normal subgroup of $\Sigma$ distinct from the identity; using I, § 5, Exercise 17, show that $\Sigma = \Delta . \Phi_c = \Delta . \Gamma_c$ and, using the fact that $\Gamma_c$ is commutative, deduce that every commutator in $\Sigma$ is contained in $\Delta$.)

(e) Deduce from (c), (d) and Exercise 13(c) that the group $\mathbf{PSL}(E)$ is simple except when $\dim E = 2$ and $K$ has 2 or 3 elements.

(f) If $\dim E = n + 1$ and $K$ is a field with $q$ elements, show that the order of $\mathbf{PGL}(E)$ is

(use (a), Exercise 9(a) of § 9, and the fact that $K$ is necessarily commutative (V, § 11, Exercise 14 and VIII, § 11, no. 1, Theorem 1)).*

(g) Show that if $\dim E = 2$ and $K$ is a field with 2 (resp. 3) elements, $\mathbf{PSL}(E)$ is isomorphic to the symmetric group $S_3$ (resp. the alternating group $A_4$). (Considering $\mathbf{PSL}(E)$ as a group of permutations of $\mathbf{P}(E)$, note that it contains every transposition (resp. every cycle $(a\ b\ c)$ and use f).)

(h) Show that, except in the cases considered in (g), every normal subgroup of $\mathbf{SL}(E)$ distinct from $\mathbf{SL}(E)$ is contained in the centre of $\mathbf{SL}(E)$ (use (d) and the fact that every transvection is a commutator in $\mathbf{SL}(E)$ (Exercise 13(c))).