Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group.

*Ideas*:

We'd be doin real good if we could find the set of element orders $\{o(A):A\in GL(n,q)\}$, so I am looking for things to help me figure this out.

1. The order of $GL(n,q)$ is $$|GL(n,q)|=\prod\_{k=0}^{n-1}\left(q^n-q^k\right)$$ The set of prime factors of this number is a little unpredictable so I'm not sure whether I can use this.
2. If $q$ is prime, the maximum order of an element in $GL(n,q)$ is at most $p^n-1$ (and this bound is strict I think). Also I guess the maximal power of $p$ that occurs as an order of this group is $p^{\lceil \log\_pn\rceil}$. ([source](https://mathoverflow.net/questions/109483/maximal-order-of-elements-in-gln-p/109488#109488))
3. I *am* aware of [this](https://math.stackexchange.com/a/79056/53776), which I believe implies that $\text{Exp}(GL(2,p^n))=\text{lcm}\{p(p^{2n}-1),p-1\}$. I am trying to think of how to extend it to the general case but I'm not sure.

And I'm stuck; can't think of anything else that might help.