I wanted to see if there is any connection between the invertibility of a matrix and the invertibility of a particular block of the matrix.
Particularly I want to find out the largest size of a subgroup of $GL_n(\mathbb{F}_q)$ with the property that the first $k \times k$ block of all the matrices in that subgroup is invertible.
I have formalized the problem (trying to do away with matrices) in the following manner.
Let $V$ be a vector space over $\mathbb{F_q}$ of dimension $n$. Let $W$ be a subspace of $V$ so that V = W \oplus W' and dim($W$) $= k$. Let $P$ denote the projection along W' onto $W$ and $G$ be a subgroup of $GL(V)$. Then define $\tilde{G} = \{ T \in G |$ $P\circ T: W \to W$ is invertible $\}$.
1) Under what conditions is $G = \tilde{G}$?
2) What is the largest size of $G$ such that $G=\tilde{G}$?
A few observations:
1) If $G = GL(V)$, then $G \neq \tilde{G}$.
2) I have a group of size $|M_{k \times (n-k)}(\mathbb{F}_q)|$ ($M_{k \times (n-k)}(\mathbb{F}_q)$ is the additive group of $k \times (n-k)$ matrices over $\mathbb{F}_q$) for which $G = \tilde{G}$ holds.
P.S: There was a power cut in the campus suddenly. I will complete the details, if required, at a later moment. Sorry for the inconvenience.
Thank you,
Isomorphism