Actions of Δ(3,n,k) on projective line

Abstract

Each conjugacy class of actions of the triangle group $\Delta (3,n,k)$ over the projective line $PL\left(F_q\right)$ can be represented by a coset diagram $D(\theta ,q)$, where $\theta \in F_q$ and $q$ is a prime number. In this paper, we have considered conjugacy classes which arise from the actions of $\Delta (3,n,k)=〈r,s:r^3=s^n={\left(rs\right)}^k=1〉$ over $PL\left(F_q\right)$, where $F_q$ is finite field. The points of $PL\left(F_q\right)$ are the elements of $F_q$ together with the additional point $\infty .$