  Let $X$ be a compact Riemann surface of genus $g\geq 2$. A cyclic subgroup of
prime order $p$ of $Aut(X)$ is called properly $(p,h)$-gonal if it has a fixed
point and the quotient surface has genus $h$. We show that if $p>6h+6$, then a
properly $(p,h)$-gonal subgroup of $Aut(X)$ is unique. We also discuss some
related results.
