On ${\text{c - 3}}$-Transitive Automorphism Groups of Cyclically Ordered Sets

Abstract

An automorphism group $G$ of a cyclically ordered set $\left\langle {X,C} \right\rangle$ is said to be ${\text{c - 3}}$-transitive if for any elements $x_i ,y_i \in {\text{X }},{\text{ }}i = 1,2,3$, such that $C(x_1 ,x_2 ,x_3 )$ and $C(y_1 ,y_2 ,y_3 )$ there exists an element $g \in G$ satisfying $g(x_i ) = y_i$, $i = 1,2,3$. We prove that if an automorphism group of a cyclically ordered set is ${\text{c - 3}}$-transitive, then it is simple. A description of ${\text{c - 3}}$-transitive automorphism groups with Abelian two-point stabilizer is given.