Suppose $G$ is a subgroup of $GL(n,q)$ given by a list of generators. What is known about the complexity of the corresponding "membership problem", that is, the problem of deciding whether a given $g \in GL(n,q)$ belongs to $G$ ?
While it would be great to have an actual bound on the complexity, or to know that some other problems reduce to this one, I would also be happy with empirical or even anecdotal evidence.
To me, it seems that the problem is very tractable when $n$ is less than 10 and $q$ is less than 50, but becomes hard very very rapidly. I think trying to compute with $GL(25, 81)$ would be a disaster. What about $n\ge 500$ and $p\ge 10,000$ ?
