Build a finite field F of order pn in the usual way. Then construct the (multiplicative) group of all invertible (nonsingular) m×m matrices over this field with the command G = GL(m, F) (``the general linear group''). What is the order of this group? In other words, find a general expression for the order of this group.
Your answer should be a function of m, p and n. Provide a complete explanation of the logic behind your solution (i.e. something resembling a proof). Also provide tests in Sage that your answer is correct.
Hints: G.order() will help you test and verify your hypotheses. Small examples in Sage (listing all the elements of the group) might aid your intuition---which is why this is a Sage exercise. Small means 2×2 and 3×3 matrices and finite fields with 2,3,4,5 elements, at most. Results do not really depend on each of p and n, but rather just on pn.
Realize this group is interesting because it contains representations of all the invertible (i.e. 1-1 and onto) linear transformations from the (finite) vector space Fm to itself.