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 center (isomorphic to the multiplicative group of the center of $K$, cf. II, p. 182, exerc. 26). b) Let $\mathbf{PSL}(E)$ denote the canonical image of the group $\mathbf{SL}(E)$ (exerc. 13) in $\mathbf{PGL}(E)$; it is a normal subgroup of $\mathbf{PGL}(E)$, containing the commutator group of $\mathbf{PGL}(E)$ when $E$ is of finite dimension. Show that $\mathbf{PSL}(E)$ is a doubly transitive group (I, p. 131, exerc. 14) of permutations of $\mathbf{P}(E)$. c) Let $a \neq 0$ be an element of $E$, and let $e = \pi(a) \in \mathbf{P}(E)$ (with the notations of II, p. 133). Show that the transvections of the form $x \mapsto x + a \varphi(x)$ (where $\varphi \in E^*$ is such that $\varphi(a) = 0$) 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 moreover that the union of the conjugate subgroups of $\Gamma_e$ in $\mathbf{PSL}(E)$ generates $\mathbf{PSL}(E)$. d) Let $\Sigma$ be a primitive group (I, p. 131, exerc. 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 conjugate subgroups of $\Gamma_c$ 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, p. 131, exerc. 17, show that one has $\Sigma = \Delta . \Phi_c = \Delta . \Gamma_c$, 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 exerc. 13 c) that the group $\mathbf{PSL}(E)$ is simple unless $\dim E = 2$ and $K$ has 2 or 3 elements. f) If $\dim E = n + 1$ and if $K$ is a field with $q$ elements, show that the order of $\mathbf{PGL}(E)$ is

(use a), exerc. 9 a) of II, p. 202, and the fact that $K$ is necessarily commutative (V, § 11, exerc. 14 and VIII, § 11, no. 1, th. 1)). g) Show that if $\dim E = 2$ and if $K$ is a field with 2 (resp. 3) elements, $\mathbf{PSL}(E)$ is isomorphic to the symmetric group $S_3$ (resp. to the alternating group $A_4$). (Considering $\mathbf{PSL}(E)$ as a permutation group of $\mathbf{P}(E)$, remark 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 center of $\mathbf{SL}(E)$ (use d) and the fact that every transvection is a commutator in $\mathbf{SL}(E)$ (exerc. 13 c))).