We know PGL(2, $q$) has elements of order $q+1$ or $q-1$. Suppose $k\neq 1$, $2$ divide $q+1$ or $q-1$. It is clear that PGL(2, $q$) has an elements of order $k$. I would like to know what is the number of the elements of order $k$ and how we can get it?
If "elementary" means not using complex numbers, consider this.


First minimize the Rayleigh ratio $R(x)=(x^TAx)/(x^Tx).$ The minimum exists and is real.
This is your first eigenvalue. 
Then you repeat the usual proof by induction in dimension of the space.
Alternatively you can consider the minimax or maximin problem with the same Rayleigh ratio,
(find the minimum of a restriction on a subspace, then maximum over all
subspaces) and it will give you all eigenvalues.


But of course any proof requires some topology. The standard proof requires Fundamental theorem of
Algebra, this proof requires existence of a minimum.
