Is it 'expected' that PSL(2,p) contains a cyclic subgroup of order $\tfrac{p+1}{2}$? I read such an element exists and generates a "nonsplit torus", but I'm not sure what that means.
Is there an easy explanation why such an element should exist and have a cycle structure $\tfrac{p+1}{2},\tfrac{p+1}{2}$ in its action on the projective line?
(Edit: On a sidenote, it is on the other hand obvious that PSL(2,p) has cyclic subgroups of order (p-1)/2 and p, generated by $(\begin{smallmatrix}r&0\\0&\tfrac{1}{r}\end{smallmatrix})$ and $(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix})$ where r generates $GF(p)^*$ )