Is there a relation between the order of an element in $SL(2,F)$ (where F is a finite field) and that of its projection down to $PSL(2,F)$ ?
For example: if $x$ has order 4 in $SL(2,F)$, its image under the canonical projection has order 2 in $PSL(2,F)$. Is it correct to say that the order of the image would be one of the prime divisors of the order of the lift ?
Thanks so much for any guide.