I'd rather write
(1) For a prime, we have that . Hint for proof: count how many inversible matrices are there by counting in how many ways can you choose the first column of such a matrix, then in how many ways can its 2nd column be chosen, etc.
2) Since is an epimorphism of groups, we get that (Hint: what is ?)
3) From the above it follows at once that is the automorphism group of a cyclic group, so...