  If $p$ is prime, a compact Riemann surface $X$ of genus $g\geq 2$ is called
cyclic $p$-gonal if it admits a cyclic group of automorphisms $C_{p}$ of order
$p$ such that the quotient space $X/C_{p}$ has genus 0. If in addition $C_{p}$
is not normal in the full automorphism $G$, then we call $G$ a non-normal
cyclic $p$-gonal group. In the following we classify all non-normal $p$-gonal
groups.
