TITLE: Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?
QUESTION [19 upvotes]: A very naive question :
I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of Cayley-Graves octaves (edit: octonions) that preserve up to sign the basis $e_i$, $i=1..7$of imaginary octaves. Does this happen for other values of $(n,q)$ (as in the title) ?

REPLY [10 votes]: There's an elementary (non-cohomological) group-theoretic explanation for why $GL(n,q)$ must split over its natural module in odd characteristic. Suppose $N \triangleleft G$, where
$G/N \cong GL(n,q)$ with $q$ odd and $N$ is isomorphic to the natural module for $G/N$. The negative of the identity matrix in $GL(n,q)$ is a central element of order $2$ that acts on $N$ with no nonidentity fixed points. Thus there exists $T \triangleleft G$ with
$N \subseteq T$ such that $T/N$ has order $2$, and if $S$ is a Sylow $2$-subgroup of $T$ then ${\bf C}_N(S) = 1$. Now let $K = {\bf N}_G(S)$. By the Frattini argument,
$G = TK = NSK = NK$. Also, $N \cap K = 1$ since $T$ must centralize $N \cap K$. Thus $K$ is the desired complement for $N$ in $G$.