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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

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    @Jyrki: I think it is usually (but not always) a maximal subgroup (not just with the specified property), since it is a stabilizer of a subspace. If n = 2k, then you also get [0,1;1,0] in the maximal subgroup, but that doesn't satisfy the property. Size-wise it is pretty big. One might be able to consult the list of maximal subgroups of GL to rule out any other possibilities.2011-09-29

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