Embeddings of $\mathrm{GL}(n-1,q)$ into $\mathrm{GL}(n,q)$ Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $\mathrm{GL}(n-1,q)$ into $\mathrm{GL}(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over $\mathrm{GF}(q)$ and $H$ is a subgroup of $\mathrm{GL}(V)$ fixing a nonzero vector in $V$. Then $H$ is isomorphic to $\mathrm{GL}(n-1,q)$.
 A: So the question should be: if $H$ is a subgroup of ${\rm GL}(n,q)$ isomorphic to ${\rm GL}(n-1,q)$, then does $H$ necessarily fix two complementary subspaces of $V$ of dimensions $1$ and $n-1$?
This is a nontrivial question. Let $K$ be the field of order $q$. Then the only simple $KH$-modules of dimension less than $n$ have dimensions 1 and $n-1$, so the constituents of $V$ as $KH$-module must have dimensions $1$ and $n-1$, but then the question is whether this module is completely reducible, and that depends on the 1-cohomology of $H$ on the $(n-1)$-dimensional constituent.
There are some instances in which there are embeddings that are not completely reducible, so the answer to the question is no. These include $n=3$, $q=2^k$ for $k \ge 2$, and also $n=4$, $q=2$. I think the answer is yes in all other cases, but I would not bet on it!
Added later: here is a rough description of the non-natural examples when $n=3$ and $q=2^k$. Consider first the subgroup of matrices of the form
$$\left( \begin{array}{ccc}a&b&0\\c&d&0\\x&y&1\end{array} \right).$$
in which $ab-cd=1$. This is a semidirect product of $K^2$ with ${\rm SL}(2,q)$ with the natural action. It turns out that $H^1({\rm SL}(2,2^k),K^2)$ is nonzero, and has dimension 1 over $K$, when $k \ge 2$.
Let $C \cong {\rm SL}(2,q)$ be the a complement of $K^2$ corresponding to a nonzero element of $H^1({\rm SL}(2,q),K^2)$. Then $C$ does not act completely reducibly on $V$, and the subgroup $H$ gerenrated by $C$ and scalar matrices is isomorphic to ${\rm GL}(2,q)$. Then (depending on whether your matrices act on the left or right) $H$ fixes a 1-dimensional or a 2-dimensional subspace of $V$, but not both, and the transpose of $H$ does the opposite. So there are just two conjugacy classes of non-natural ${\rm GL}(2,q)$s in ${\rm GL}(3,q)$.
A: If your last sentence is you definition of "natural", then the answer is that other embeddings are possible. Since the ones below follow a simple pattern, you will probably want to extend "natural" to include them as well.
You can always embed $\mathbf{GL}(n-1,F)$ into $\mathbf{GL}(n,F)$ by sending $A$ to the block matrix
$$
\begin{pmatrix} A&0\\0&\det A\end{pmatrix}.
$$
Unless $F=\mathbf{GF}(2)$, the image of this embedding does not have any fixed vectors.
